Statistics

In the world of machine learning, evaluating the performance of a model is just as important as building the model itself. One of the most fundamental tools for this purpose is the confusion matrix. This powerful yet simple concept helps data scientists and machine learning practitioners assess the accuracy of classification algorithms, providing insights into how well a model is performing in predicting various classes.

In this blog, we will explore the concept of a confusion matrix using a spam email example. We highlight the 4 key metrics you must understand and work on while working with a confusion matrix.

What is a Confusion Matrix?

A confusion matrix is a table that is used to describe the performance of a classification model. It compares the actual target values with those predicted by the model. This comparison is done across all classes in the dataset, giving a detailed breakdown of how well the model is performing. 

Here’s a simple layout of a confusion matrix for a binary classification problem:

In a binary classification problem, the confusion matrix consists of four key components: 

1. True Positive (TP): The number of instances where the model correctly predicted the positive class. 
2. False Positive (FP): The number of instances where the model incorrectly predicted the positive class when it was actually negative. Also known as Type I error. 
3. False Negative (FN): The number of instances where the model incorrectly predicted the negative class when it was actually positive. Also known as Type II error. 
4. True Negative (TN): The number of instances where the model correctly predicted the negative class.

Why is the Confusion Matrix Important?

The confusion matrix provides a more nuanced view of a model’s performance than a single accuracy score. It allows you to see not just how many predictions were correct, but also where the model is making errors, and what kind of errors are occurring. This information is critical for improving model performance, especially in cases where certain types of errors are more costly than others. 

For example, in medical diagnosis, a false negative (where the model fails to identify a disease) could be far more serious than a false positive. In such cases, the confusion matrix helps in understanding these errors and guiding the development of models that minimize the most critical types of errors.

Also learn about the Random Forest Algorithm and its uses in ML

Scenario: Email Spam Classification

Suppose you have built a machine learning model to classify emails as either “Spam” or “Not Spam.” You test your model on a dataset of 100 emails, and the actual and predicted classifications are compared. Here’s how the results could break down: 

• Total emails: 100 
• Actual Spam emails: 40 
• Actual Not Spam emails: 60

After running your model, the results are as follows: 

• Correctly predicted Spam emails (True Positives, TP): 35
• Incorrectly predicted Spam emails (False Positives, FP): 10

• Incorrectly predicted Not Spam emails (False Negatives, FN): 5
• Correctly predicted Not Spam emails (True Negatives, TN): 50

Understanding 4 Key Metrics Derived from the Confusion Matrix

The confusion matrix serves as the foundation for several important metrics that are used to evaluate the performance of a classification model. These include:

1. Accuracy

• Formula for Accuracy in a Confusion Matrix:

Explanation: Accuracy measures the overall correctness of the model by dividing the sum of true positives and true negatives by the total number of predictions.

• Calculation for accuracy in the given confusion matrix:

This equates to = 0.85 (or 85%). It means that the model correctly predicted 85% of the emails.

2. Precision

• Formula for Precision in a Confusion Matrix:

Explanation: Precision (also known as positive predictive value) is the ratio of correctly predicted positive observations to the total predicted positives.

It answers the question: Of all the positive predictions, how many were actually correct?

• Calculation for precision of the given confusion matrix

It equates to ≈ 0.78 (or 78%) which highlights that of all the emails predicted as Spam, 78% were actually Spam.

3. Recall (Sensitivity or True Positive Rate)

• Formula for Recall in a Confusion Matrix

Explanation: Recall measures the model’s ability to correctly identify all positive instances. It answers the question: Of all the actual positives, how many did the model correctly predict?

• Calculation for recall in the given confusion matrix

It equates to = 0.875 (or 87.5%), highlighting that the model correctly identified 87.5% of the actual Spam emails.

4. F1 Score

• F1 Score Formula:

Explanation: The F1 score is the harmonic mean of precision and recall. It is especially useful when the class distribution is imbalanced, as it balances the two metrics.

• F1 Calculation:

This calculation equates to ≈ 0.82 (or 82%). It indicates that the F1 score balances Precision and Recall, providing a single metric for performance.

Understand the basics of Binomial Distribution and its importance in ML

Interpreting the Key Metrics

• High Recall: The model is good at identifying actual Spam emails (high Recall of 87.5%). 

• Moderate Precision: However, it also incorrectly labels some Not Spam emails as Spam (Precision of 78%). 

• Balanced Accuracy: The overall accuracy is 85%, meaning the model performs well, but there is room for improvement in reducing false positives and false negatives. 

• Solid F1 Score: The F1 Score of 82% reflects a good balance between Precision and Recall, meaning the model is reasonably effective at identifying true positives without generating too many false positives. This balanced metric is particularly valuable in evaluating the model’s performance in situations where both false positives and false negatives are important.

Conclusion

The confusion matrix is an indispensable tool in the evaluation of classification models. By breaking down the performance into detailed components, it provides a deeper understanding of how well the model is performing, highlighting both strengths and weaknesses. Whether you are a beginner or an experienced data scientist, mastering the confusion matrix is essential for building effective and reliable machine learning models.

In the domain of machine learning, evaluating the performance and results of a classification model is a mandatory step. There are numerous metrics available to get this done.

The ones discussed in this blog are the AUC (Area Under the Curve) and ROC (Receiver Operating Characteristic). It stands out for its effectiveness in measuring the performance of classification models and multi-class classification problems.

The Confusion Matrix

Before diving into the details of the metric AUC-ROC curve, it is imperative that an understanding of the confusion matrix is developed beforehand.

The confusion matrix is a key tool used in the process of evaluating the performance of classification models, it is essentially a table that summarizes the outcomes of the model’s predictions against the actual outcomes – in short, it shows how the model is confused.

The confusion matrix is represented as follows:

1. True Positives (TP): Instances where the model correctly predicted the positive class.
2. True Negatives (TN): Instances where the model correctly predicted the negative class.
3. False Positives (FP): Instances where the model incorrectly predicted the positive class.
4. False Negatives (FN): Instances where the model incorrectly predicted the negative class.

Using this confusion matrix, many metrics are derived, and it is also used in the calculation of AUC-ROC which will now be discussed.

Learn in detail about what is a confusion matrix

What is ROC?

The Receiver Operating Characteristic (ROC) curve is a visual and graphical representation of a classification model’s performance across various thresholds. It is created by first calculating the True Positive Rate (TPR) and False Positive Rate (FPR) at every threshold value and then plotting them against each other.

An ideal model will have a TPR of 1 and an FPR of 0 which ultimately means it did not misclassify any instance in the data and there is perfect separation between the positive and the negative class. A visual representation of such an ideal classification model is shown below:

The true positive rate or TPR is calculated as:

The false positive rate or FPR is calculated as:

Example

Consider a hypothetical example of a medical test to detect a certain disease. There are 100 patients and 40 of these patients have the disease. We will use this example to create the ROC curve to have an idea other than the ideal model.

Consider that our classification model performed as such:

The calculation of TPR and FPR is carried out as:

The ROC curve will then look something like this:

This plot is plotted using Python and the code for it is as follows:

What is AUC?

The Area Under the Curve (AUC) is also sometimes referred to as the area under the ROC curve. It is a single scalar value that summarizes the performance of a classification model across all threshold values.

It represents the likelihood that the model will rank a randomly chosen positive instance higher than a randomly chosen negative instance.

The ideal model as shown in the ROC curve image has an AUC of 1, which means that there is a 100% probability that the model will correctly rank a randomly chosen positive instance higher than a randomly chosen negative instance.

AUC Interpretation

To understand AUC more intuitively, consider a medical test carried out to detect a disease. An AUC score of 1 would mean that the medical test perfectly separates the healthy and diseased individuals, always giving a higher score to those with the disease.

On the other hand, an AUC score of 0.5 is not better than random guessing or a random chance model that only correctly ranks individuals with the disease half of the time.

In summary:

• AUC = 1: Perfect model.
• 0.5 < AUC < 1: Good model, better than random chance.
• AUC = 0.5: Model with no discrimination capability.
• AUC < 0.5: Model worse than random chance.

Example with Real-World Data

Consider a dataset with two features, namely “Age” and “Income” and a binary target variable “Default” which shows whether a customer defaults on a loan or not.

Model Predictions

Suppose now that we train a logistic regression model and obtain the following probabilities for the “Default” target variable:

ROC and AUC Calculation

Finally, we plot the TPR against the FPR at different threshold values to create an ROC curve and calculate the AUC score of our model.

From our logistic regression model, we get an AUC of 0.94, indicating near-perfect model performance.

The code for the plot is mentioned below.

Note: Achieving near-perfect model performance is often unrealistic in real-world situations. The high score we saw is probably a result of the small sample size, which might not accurately reflect real-life situations.

Practical Applications

Model Evaluation

The ROC curve and the AUC score are widely used in the evaluation of the performance of classification models, especially when dealing with imbalance datasets.

By understanding and examining the trade-offs between TPR and FPR and identifying which is more relevant to the problem at hand, one can choose the optimal threshold to maximize the model’s effectiveness.

Read more about classification using decision trees

Threshold Selection

In practice, ROC curves greatly help in the selection of the optimal threshold for classification problems. For example, in medical diagnostics, one might choose a threshold that minimizes false negatives to ensure that fewer instances of the disease are missed, even if it means accepting a higher false positive rate.

Comparing Models

AUC is also a great measure for comparing the performance of two different models. The model with the greater area under the curve or in other words, having a higher AUC is generally preferred as it indicates better overall model performance.

Conclusion

Understanding the nuances of the ROC curve and the AUC score is essential for evaluating the performance of classification models.

These metrics provide a comprehensive picture of the trade-offs between true positive rates and false positive rates at different threshold values, effectively helping data scientists and practitioners make informed decisions about their models.

In the vast forest of machine learning algorithms, one algorithm stands tall like a sturdy tree – Random Forest. It’s an ensemble learning method that’s both powerful and flexible, widely used for classification and regression tasks.

But what makes the random forest algorithm so effective? How does it work?

In this blog, we’ll explore the inner workings of Random Forest, its advantages, limitations, and practical applications. 

What is a Random Forest Algorithm? 

Imagine a dense forest with numerous trees, each offering a different path to follow. Random Forest Algorithm is like that: an ensemble of decision trees working together to make more accurate predictions.

By combining the results of multiple trees, the algorithm improves the overall model performance, reducing errors and variance.

Why the Name ‘Random Forest’? 

The name “Random Forest” comes from the combination of two key concepts: randomness and forests. The “random” part refers to the random selection of data samples and features during the construction of each tree, while the “forest” part refers to the ensemble of decision trees.

This randomness is what makes the algorithm robust and less prone to overfitting. 

Common Use Cases of Random Forest Algorithm

Random Forest Algorithm is highly versatile and is used in various applications such as: 

• Classification: Spam detection, disease prediction, customer segmentation. 

• Regression: Predicting stock prices, house values, and customer lifetime value. 

Also learn about Linear vs Logistic regression

Understanding the Basics 

Before diving into Random Forest, let’s quickly revisit the concept of Decision Trees. 

Decision Trees Recap 

A decision tree is a flowchart-like structure where internal nodes represent decisions based on features, branches represent the outcomes of these decisions, and leaf nodes represent final predictions. While decision trees are easy to understand and interpret, they can be prone to overfitting, especially when the tree is deep and complex.

Key Concepts in Random Forest 

• Ensemble Learning: This technique combines multiple models to improve performance. Random Forest is an example of ensemble learning where multiple decision trees work together to produce a more accurate and stable prediction. 

Read in detail about ensemble methods in machine learning

• Bagging (Bootstrap Aggregating): In Random Forest, the algorithm creates multiple subsets of the original dataset by sampling with replacement (bootstrapping). Each tree is trained on a different subset, which helps in reducing variance and preventing overfitting. 

• Feature Randomness: During the construction of each tree, Random Forest randomly selects a subset of features to consider at each split. This ensures that the trees are diverse and reduces the likelihood that a few strong predictors dominate the model.

How Does Random Forest Work? 

Let’s break down the process into two main phases: training and prediction. 

Training Phase 

• Creating Bootstrapped Datasets: The algorithm starts by creating multiple bootstrapped datasets by randomly sampling the original data with replacement. This means some data points may be repeated, while others may be left out. 

• Building Multiple Decision Trees: For each bootstrapped dataset, a decision tree is constructed. However, instead of considering all features at each split, the algorithm randomly selects a subset of features. This randomness ensures that the trees are different from each other, leading to a more generalized model. 

Prediction Phase 

• Voting in Classification: When it’s time to make predictions, each tree in the forest casts a vote for the class label. The final prediction is determined by the majority vote among the trees. 

• Averaging in Regression: For regression tasks, instead of voting, the predictions from all the trees are averaged to get the result.

Another interesting read: Sustainability Data and Machine Learning

Advantages of Random Forest 

Random Forest is popular for good reasons. Some of these include:

High Accuracy 

By aggregating the predictions of multiple trees, Random Forest often achieves higher accuracy than individual decision trees. The ensemble approach reduces the impact of noisy data and avoids overfitting, making the model more reliable. 

Robustness to Overfitting 

Overfitting occurs when a model performs well on the training data but poorly on unseen data. Random Forest combats overfitting by averaging the predictions of multiple trees, each trained on different parts of the data. This ensemble approach helps the model generalize better. 

Handles Missing Data 

Random Forest can handle missing values naturally by using the split with the majority of the data and by averaging the outputs from trees trained on different parts of the data. 

Feature Importance 

One of the perks of Random Forest is its ability to measure the importance of each feature in making predictions. This is done by evaluating the impact of each feature on the model’s performance, providing insights into which features are most influential.

Limitations of Random Forest 

While Random Forest is a powerful tool, it’s not without its drawbacks. A few limitations associated with random forests are:

Computational Cost 

Training multiple decision trees can be computationally expensive, especially with large datasets and a high number of trees. The algorithm’s complexity increases with the number of trees and the depth of each tree, leading to longer training times. 

Interpretability 

While decision trees are easy to interpret, Random Forest, being an ensemble of many trees, is more complex and harder to interpret. The lack of transparency can be a disadvantage in situations where model interpretability is crucial. 

Bias-Variance Trade-off 

Random Forest does a good job managing the bias-variance trade-off, but it’s not immune to it. If not carefully tuned, the model can still suffer from bias or variance issues, though typically less so than a single decision tree. 

Hyperparameter Tuning in Random Forest

While we understand the benefits and limitations of Random Forest, let’s take a deeper look into working with the algorithm. Understanding and working with relevant hyperparameters is a crucial part of the process.

It is an important aspect because tuning the hyperparameters of a Random Forest can significantly impact its performance. Here are some key hyperparameters to consider:

Master hyperparameter tuning for machine learning models

Key Hyperparameters 

• Number of Trees (n_estimators): The number of trees in the forest. Increasing this generally improves performance but with diminishing returns and increased computational cost. 

• Maximum Depth (max_depth): The maximum depth of each tree. Limiting the depth can help prevent overfitting. 

• Number of Features (max_features): The number of features to consider when looking for the best split. Lower values increase diversity among trees but can also lead to underfitting. 

Techniques for Tuning 

• Grid Search: This exhaustive search technique tries every combination of hyperparameters within a specified range to find the best combination. While thorough, it can be time-consuming. 

• Random Search: Instead of trying every combination, Random Search randomly selects combinations of hyperparameters. It’s faster than Grid Search and often finds good results with less computational effort. 

• Cross-Validation: Cross-validation is essential in hyperparameter tuning. It splits the data into several subsets and uses different combinations for training and validation, ensuring that the model’s performance is not dependent on a specific subset of data.

Practical Implementation 

To understand how Random Forest works in practice, let’s look at a step-by-step implementation using Python. 

Setting Up the Environment 

You’ll need the following Python libraries: scikit-learn for the Random Forest implementation, pandas for data handling, and numpy for numerical operations.

Example Dataset 

For this example, we’ll use the famous Iris dataset, a simple yet effective dataset for demonstrating classification algorithms. 

Step-by-Step Code Walkthrough 

• Data Preprocessing: Start by loading the data and handling any missing values, though the Iris dataset is clean and ready to use.

• Training the Random Forest Model: Instantiate the Random Forest classifier and fit it to the training data. 

• Evaluating the Model: Use the test data to evaluate the model’s performance.

• Hyperparameter Tuning: Use Grid Search or Random Search to find the optimal hyperparameters.

Comparing Random Forest with Other Algorithms 

Random Forest vs. Decision Trees 

While a single decision tree is easy to interpret, it’s prone to overfitting, especially with complex data. Random Forest reduces overfitting by averaging the predictions of multiple trees, leading to better generalization.

Explore the boosting algorithms used to enhance ML model accuracy

Random Forest vs. Gradient Boosting 

Both are ensemble methods, but they differ in approach. Random Forest builds trees independently, while Gradient Boosting builds trees sequentially, where each tree corrects the errors of the previous one. Gradient Boosting often achieves better accuracy but at the cost of higher computational complexity and longer training times. 

Random Forest vs. Support Vector Machines (SVM) 

SVMs are powerful for high-dimensional data, especially when the number of features exceeds the number of samples. However, SVMs are less interpretable and more sensitive to parameter tuning compared to Random Forest. Random Forest tends to be more robust and easier to use out of the box.

Explore the Impact of Random Forest Algorithm

Random Forest is a powerful and versatile algorithm, capable of handling complex datasets with high accuracy. Its ensemble nature makes it robust against overfitting and capable of providing valuable insights into feature importance. 

As you venture into the world of machine learning, remember that a well-tuned Random Forest can be the key to unlocking insights hidden deep within your data. Keep experimenting, stay curious, and let your models grow as robust as the forest itself!

In the realm of statistics and machine learning, understanding various probability distributions is paramount. One such fundamental distribution is the Binomial Distribution.

This distribution is not only a cornerstone in probability theory but also plays a crucial role in various machine learning algorithms and applications.

In this blog, we will delve into the concept of binomial distribution, its mathematical formulation, and its significance in the field of machine learning.

What is Binomial Distribution?

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent and identically distributed Bernoulli trials.

A Bernoulli trial is a random experiment where there are only two possible outcomes:

• success (with probability ( p ))
• failure (with probability ( 1 – p ))

Mathematical Formulation

The probability of observing exactly k successes in n trials is given by the binomial probability formula:

Example 1: Tossing One Coin

Let’s start with a simple example of tossing a single coin.

Parameters

• Number of trials (n) = 1
• Probability of heads (p) = 0.5
• Number of heads (k) = 1

Calculation

• Binomial coefficient

• Probability

So, the probability of getting exactly one head in one toss of a coin is 0.5 or 50%.

Example 2: Tossing Two Coins

Now, let’s consider the case of tossing two coins.

Parameters

• Number of trials (n) = 2
• Probability of heads (p) = 0.5
• Number of heads (k) = varies (0, 1, or 2)

Calculation for k = 0

• Binomial coefficient

• Probability

P(X = 0) = 1 × (0.5)⁰ × (1 – 0.5)² = 1 × 1 × 0.25 = 0.25

Calculation for k = 1

• Binomial coefficient

• Probability

P(X = 1) = 1 × (0.5)¹ × (1 – 0.5)¹ = 2 × 0.5 × 0.5 = 0.5

Calculation for k = 2

• Binomial coefficient

• Probability

P(X = 2) = 1 × (0.5)² × (1 – 0.5)⁰ = 1 × 0.25 × 1 = 0.25

So, the probabilities for different numbers of heads in two-coin tosses are:

• P(X = 0) = 0.25 – no heads
• P(X = 1) = 0.5 – one head
• P(X = 2) = 0.25 – two heads

Detailed Example: Predicting Machine Failure

Let’s consider a more practical example involving predictive maintenance in an industrial setting. Suppose we have a machine that is known to fail with a probability of 0.05 during a daily checkup. We want to determine the probability of the machine failing exactly 3 times in 20 days.

Step-by-Step Calculation

1. Identify Parameters

• Number of trials (n) = 20 days
• Probability of success (p) = 0.05 – failure is considered a success in this context
• Number of successes (k) = 3 failures

2. Apply the Formula

3. Compute Binomial Coefficient

4. Calculate Probability

• Plugging the values into the binomial formula

• Substitute the values

P(X = 3) = 1140 × (0.05)³ × (0.95)¹⁷

• Calculate (0.05)³

(0.05)³ = 0.000125

• Calculate (0.95)¹⁷

(0.95)¹⁷ ≈ 0.411

5. Multiply all Components Together

P(X = 3) = 1140 × 0.000125 × 0.411 ≈ 0.0585

Therefore, the probability of the machine failing exactly 3 times in 20 days is approximately 0.0585 or 5.85%. 

Role of Binomial Distribution in Machine Learning 

The binomial distribution is integral to several aspects of machine learning, providing a foundation for understanding and modeling binary events, hypothesis testing, and beyond.

Let’s explore how it intersects with various machine-learning concepts and techniques. 

Binary Classification

In binary classification problems, where the outcomes are often categorized as success or failure, the binomial distribution forms the underlying probabilistic model. For instance, if we are predicting whether an email is spam or not, each email can be thought of as a Bernoulli trial.

Algorithms like Logistic Regression and Support Vector Machines (SVM) are particularly designed to handle these binary outcomes. 

Understanding the binomial distribution helps in correctly interpreting the results of these classifiers. The performance metrics such as accuracy, precision, recall, and F1-score ultimately derive from the binomial probability model.

This understanding ensures that we can make informed decisions about model improvements and performance evaluation.

Hypothesis Testing

Statistical hypothesis testing, essential in validating machine learning models, often employs the binomial distribution to ascertain the significance of observed outcomes.

For instance, in A/B testing, which is widely used in machine learning for comparing model performance or feature impact, the binomial distribution helps in calculating p-values and confidence intervals.

You can also explore an ethical way of A/B testing

Consider an example where we want to determine if a new feature in a recommendation system improves user click-through rates. By modeling the click events as a binomial distribution, we can perform a hypothesis test to evaluate if the observed improvement is statistically significant or just due to random chance.

Generative Models

Generative models such as Naive Bayes leverage binomial distributions to model the probability of observing certain classes given specific features. This is particularly useful when dealing with binary or categorical data.

In text classification tasks, for example, the presence or absence of certain words (features) in a document can be modeled using binomial distributions to predict the document’s category (class). 

By understanding the binomial distribution, we can better grasp how these models work under the hood, leading to more effective feature engineering and model tuning. 

Also explore 7 different types of statistical distributions

Monte Carlo Simulations

Monte Carlo simulations, which are used in various machine learning applications for uncertainty estimation and decision-making, often rely on binomial distributions to model and simulate binary events over numerous trials.

These simulations can help in understanding the variability and uncertainty in model predictions, providing a robust framework for decision-making in the presence of randomness.

Practical Applications in Machine Learning 

Quality Control in Manufacturing

In manufacturing, maintaining high-quality standards is crucial. Machine learning models are often deployed to predict the likelihood of defects in products.

Here, the binomial distribution is used to model the number of defective items in a batch. By understanding the distribution, we can set appropriate thresholds and confidence intervals to decide when to take corrective actions.

Explore Locust – a tool for quality assurance

Medical Diagnosis

In medical diagnosis, machine learning models assist in predicting the presence or absence of a disease based on patient data. The binomial distribution provides a framework for understanding the probabilities of correct and incorrect diagnoses.

This is critical for evaluating the performance of diagnostic models and ensuring they meet the necessary accuracy and reliability standards. 

Fraud Detection

Fraud detection systems in finance and e-commerce rely heavily on binary classification models to distinguish between legitimate and fraudulent transactions. The binomial distribution aids in modeling the occurrence of fraud and helps in setting detection thresholds that balance false positives and false negatives effectively.

Learn how cybersecurity has revolutionized with the use of data science

Customer Churn Prediction

Predicting customer churn is vital for businesses to retain their customer base. Machine learning models predict whether a customer will leave (churn) or stay (retain). The binomial distribution helps in understanding the probabilities of churn events and in setting up retention strategies based on these probabilities. 

Why Use Binomial Distribution?

Binomial distribution is a fundamental concept that finds extensive application in machine learning. From binary classification to hypothesis testing and generative models, understanding and leveraging this distribution can significantly enhance the performance and interpretability of machine learning models.

By mastering the binomial distribution, you equip yourself with a powerful tool for tackling a wide range of problems in statistics and machine learning. 

Feel free to dive deeper into this topic, experiment with different values, and explore the fascinating world of probability distributions in machine learning! 

In the world of data analysis, drawing insights from a limited dataset can often be challenging. Traditional statistical methods sometimes fall short when it comes to deriving reliable estimates, especially with small or skewed datasets. This is where bootstrap sampling, a powerful and versatile statistical technique, comes into play.

In this blog, we’ll explore what bootstrap sampling is, how it works, and its various applications in the field of data analysis.

What is Bootstrap Sampling?

Bootstrap sampling is a resampling method that involves repeatedly drawing samples from a dataset with replacements to estimate the sampling distribution of a statistic.

Essentially, you take multiple random samples from your original data, calculate the desired statistic for each sample, and use these results to infer properties about the population from which the original data was drawn.

Learn about boosting algorithms in machine learning

Why do we Need Bootstrap Sampling?

This is a fundamental question I’ve seen machine learning enthusiasts grapple with. What is the point of bootstrap sampling? Where can you use it? Let me take an example to explain this. 

Let’s say we want to find the mean height of all the students in a school (which has a total population of 1,000). So, how can we perform this task? 

One approach is to measure the height of a random sample of students and then compute the mean height. I’ve illustrated this process below.

Traditional Approach

1. Draw a random sample of 30 students from the school. 
2. Measure the heights of these 30 students. 
3. Compute the mean height of this sample. 

However, this approach has limitations. The mean height calculated from this single sample might not be a reliable estimate of the population mean due to sampling variability. If we draw a different sample of 30 students, we might get a different mean height.

Another interesting read: Machine Learning techniques

To address this, we need a way to assess the variability of our estimate and improve its accuracy. This is where bootstrap sampling comes into play.

Bootstrap Approach

1. Draw a random sample of 30 students from the school and measure their heights. This is your original sample. 
2. From this original sample, create many new samples (bootstrap samples) by randomly selecting students with replacements. For instance, generate 1,000 bootstrap samples. 
3. For each bootstrap sample, calculate the mean height. 
4. Use the distribution of these 1,000 bootstrap means to estimate the mean height of the population and to assess the variability of your estimate.

Implementation in Python

To illustrate the power of bootstrap sampling, let’s calculate a 95% confidence interval for the mean height of students in a school using Python. We will break down the process into clear steps.

Step 1: Import Necessary Libraries

First, we need to import the necessary libraries. We’ll use `numpy` for numerical operations and `matplotlib` for visualization.

Step 2: Create the Original Sample

We will create a sample dataset of heights. In a real-world scenario, this would be your collected data.

Step 3: Define the Bootstrap Function

We define a function that generates bootstrap samples and calculates the mean for each sample. 

• data: The original sample. 
• n_iterations: Number of bootstrap samples to generate. 
• -bootstrap_means: List to store the mean of each bootstrap sample. 
• -n_size: The original sample’s size will be the same for each bootstrap sample. 
• -np.random.choice: Randomly select elements from the original sample with replacements to create a bootstrap sample. 
• -sample_mean: Mean of the bootstrap sample.

Explore the use of Gini Index and Entropy in data analytics

Step 4: Generate Bootstrap Samples

We use the function to generate 1,000 bootstrap samples and calculate the mean for each.

Step 5: Calculate the Confidence Interval

We calculate the 95% confidence interval from the bootstrap means.

• np.percentile: Computes the specified percentile (2.5th and 97.5th) of the bootstrap means to determine the confidence interval.

Step 6: Visualize the Bootstrap Means

Finally, we can visualize the distribution of bootstrap means and the confidence interval. 

• plt.hist: Plots the histogram of bootstrap means. 
• plt.axvline: Draws vertical lines for the confidence interval.

By following these steps, you can use bootstrap sampling to estimate the mean height of a population and assess the variability of your estimate. This method is simple yet powerful, making it a valuable tool in statistical analysis and data science.

Read about ensemble methods in machine learning

Applications of Bootstrap Sampling

Bootstrap sampling is widely used across various fields, including the following:

Economics

Bootstrap sampling is a versatile tool in economics. It excels in handling non-normal data, commonly found in economic datasets. Key applications include constructing confidence intervals for complex estimators, performing hypothesis tests without parametric assumptions, evaluating model performance, and assessing financial risk.

For instance, economists use bootstrap to estimate income inequality measures, analyze macroeconomic time series, and evaluate the impact of economic policies. The technique is also used to estimate economic indicators, such as inflation rates or GDP growth, where traditional methods might be inadequate.

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Medicine

Bootstrap sampling is applied in medicine to analyze clinical trial data, estimate treatment effects, and assess diagnostic test accuracy. It helps in constructing confidence intervals for treatment effects, evaluating the performance of different diagnostic tests, and identifying potential confounders.

Bootstrap can be used to estimate survival probabilities in survival analysis and to assess the reliability of medical imaging techniques. It is also suitable to assess the reliability of clinical trial results, especially when sample sizes are small or the data is not normally distributed.

Also look at: Healthcare Data Exploration in Tableau

Machine Learning

In machine learning, bootstrap estimates model uncertainty, improves model generalization, and selects optimal hyperparameters. It aids in tasks like constructing confidence intervals for model predictions, assessing the stability of machine learning models, and performing feature selection.

Bootstrap can create multiple bootstrap samples for training and evaluating different models, helping to identify the best-performing model and prevent overfitting. For instance, it can evaluate the performance of predictive models through techniques like bootstrapped cross-validation.

Ecology

Ecologists utilize bootstrap sampling to estimate population parameters, assess species diversity, and analyze ecological relationships. It helps in constructing confidence intervals for population means, medians, or quantiles, estimating species richness, and evaluating the impact of environmental factors on ecological communities.

Bootstrap is also employed in community ecology to compare species diversity between different habitats or time periods.

Advantages and Disadvantages

To Sum it Up 

Bootstrap sampling is a powerful tool for data analysis, offering flexibility and practicality in a wide range of applications. By repeatedly resampling from your dataset and calculating the desired statistic, you can gain insights into the variability and reliability of your estimates, even when traditional methods fall short.

Whether you’re working in economics, medicine, machine learning, or ecology, understanding and utilizing bootstrap sampling can enhance your analytical capabilities and lead to more robust conclusions.

In data science and machine learning, decision trees are powerful models for both classification and regression tasks. They follow a top-down greedy approach to select the best feature for each split. Two fundamental metrics determine the best split at each node – Gini Index and Entropy.

This blog will explore what these metrics are, and how they are used with the help of an example.

What is the Gini Index?

It is a measure of impurity (non-homogeneity) widely used in decision trees. It aims to measure the probability of misclassifying a randomly chosen element from the dataset. The greater the value of the Gini Index, the greater the chances of having misclassifications.

Formula and Calculation

The Gini Index is calculated using the formula:

where p( j | t ) is the relative frequency of class j at node t.

• The maximum value is (1 – 1/n) indicating that n classes are equally distributed.
• The minimum value is 0 indicating that all records belong to a single class.

Another interesting read: Data Science Lifecycle

Example

Consider the following dataset.

This is also the initial root node of the decision tree, with the Gini Index as:

This result shows that the root node has maximum impurity i.e., the records are equally distributed among all output classes.

Gini Split

It determines the best feature to use for splitting at each node. It is calculated by taking a weighted sum of the Gini impurities (index) of the sub-nodes created by the split. The feature with the lowest Gini Split value is selected for splitting of the node.

Formula and Calculation

The Gini Split is calculated using the formula:

where

• ni represents the number of records at child/sub-node i.
• n represents the number of records at node p (parent-node).

Also explore: Statistical Foundations of Data Science

Example

Using the same dataset, we will determine which feature to use to perform the next split.

• For the feature “Color”, there are two sub-nodes as there are two unique values to split the data with:

• For the feature “Size”, the case is similar as that of the feature “Color”, i.e., there are also two sub-nodes when we split the data using “Size”:

Since the Gini Split for the feature “Size” is less, this is the best feature to select for this split.

What is Entropy?

Entropy is another measure of impurity, and it is used to quantify the state of disorder, randomness, or uncertainty within a set of data. In the context of decision trees, like the Gini Index, it helps in determining how a node should be split to result in sub-nodes that are as pure (homogenous) as possible.

Give it a read too: Random Forest Algorithm

Formula and Calculation

The Entropy of a node is calculated using the formula:

where p( j | t ) is the relative frequency of class j at node t.

• The maximum value is log₂(n) which indicates high uncertainty i.e., n classes are equally distributed.
• The minimum value is 0 which indicates low uncertainty i.e., all records belong to a single class.

Explore the Key Boosting Algorithms in ML and Their Applications

Example

Using the same dataset and table as discussed in the example of the Gini Index, we can calculate the Entropy (impurity) of the root node as:

This result is the same as the results obtained in the Gini Index example i.e., the root node has maximum impurity.

Information Gain

Information Gain’s objective is similar to that of the Gini Split – it aims to determine the best feature for splitting the data at each node. It does this by calculating the reduction in entropy after a node is split into sub-nodes using a particular feature. The feature with the highest information gain is chosen for the node.

Formula and Calculation

The Information Gain is calculated using the formula:

Information Gain = Entropy(Parent Node) – Average Entropy(Children)

where

• ni represents the number of records at child/sub-node i.
• n represents the number of records at the parent node.

Another useful read: 9 Important Plots in Data Science

Example

Using the same dataset, we will determine which feature to use to perform the next split:

• For the feature “Color”

• For feature “Size”:

Since the Information Gain of the split using the feature “Size” is high, this feature is the best to select at this node to perform splitting.

Gini Index vs. Entropy

Both metrics are used to determine the best splits in decision trees, but they have some differences:

• The Gini Index is computationally simpler and faster to calculate because it is a linear metric.
• Entropy considers the distribution of data more comprehensively, but it can be more computationally intensive because it is a logarithmic measure.

Use Cases

• The Gini Index is often preferred in practical implementations of decision trees due to its simplicity and speed.
• Entropy is more commonly used in theoretical discussions and algorithms like C4.5 and ID3.

Applications in Machine Learning

Decision Trees

Gini Index and Entropy are used widely in decision tree algorithms to select the best feature for splitting the data at each node/level of the decision tree. This helps improve accuracy by selecting and creating more homogeneous and pure sub-nodes.

Random Forests

Random forest algorithms, which are ensembles of decision trees, also use these metrics to improve accuracy and reduce overfitting by determining optimal splits across different trees.

Feature Selection

Both metrics also help in feature selection as they help identify features that provide the most impurity reduction, or in other words, the most information gain, which leads to more efficient and effective models.

Learn more about the different Ensemble Methods in Machine Learning

Practical Examples

1. Spam Detection
2. Customer Segmentation
3. Medical Diagnosis
4. And many more

The Final Word

Understanding the Gini Index and Entropy metrics is crucial for data scientists and anyone working with decision trees and related algorithms in machine learning. These metrics provide aid in creating splits that lead to more accurate and efficient models by selecting the optimal feature for splitting at each node.

While the Gini Index is often preferred in practice due to its simplicity and speed, Entropy provides a more detailed understanding of the data distribution. Choosing the appropriate metric depends on the specific requirements and details of your problem and machine learning task.

“Statistics is the grammar of science”, Karl Pearson.

A strong grasp of statistical concepts is crucial for anyone working with data. Whether you’re a data scientist, analyst, or researcher, understanding these fundamental principles helps you interpret data accurately, identify patterns, and make informed decisions.

From probability distributions to hypothesis testing, statistical concepts are the foundation of data analysis and machine learning.

In this blog, we’ll break down the most important statistical concepts, explaining them in simple terms with practical examples. By the end, you’ll have a solid foundation to apply statistics confidently in real-world scenarios. Let’s dive in!

 10 Statistical Concepts You Should Know

1. Descriptive Statistics:

Starting with one of the most fundamental and essential statistical concepts, descriptive statistics. Descriptive statistics are the specific methods and measures that describe the data. It’s like the foundation of your building. It is a sturdy groundwork upon which further analysis can be constructed.

Descriptive statistics can be broken down into measures of central tendency and measures of variability. 

• Measure of Central Tendency: 

Central Tendency is defined as “the number used to represent the center or middle of a set of data values”. It is a single value that is typically representative of the whole data. They help us understand where the “average” or “central” point lies amidst a collection of data points.

There are a few techniques to find the central tendency of the data, namely “Mean” (average), “Median” (middle value when data is sorted), and “Mode” (most frequently occurring values).

• Measures of variability: 

Measures of variability describe the spread, dispersion, and deviation of the data. In essence, they tell us how much each value point deviates from the central tendency. A few measures of variability are “Range”, “Variance”, “Standard Deviation”, and “Quartile Range”.

These provide valuable insights into the degree of variability or uniformity in the data. 

 2. Inferential Statistics:

Inferential statistics enable us to draw conclusions about the population from a sample of the population. Imagine having to decide whether a medicinal drug is good or bad for the general public. It is practically impossible to test it on every single member of the population.

This is where inferential statistics comes in handy. Inferential statistics employ techniques such as hypothesis testing and regression analysis (also discussed later) to determine the likelihood of observed patterns occurring by chance and to estimate population parameters.

Explore the difference between linear-regression-vs-logistic-regression

This invaluable tool empowers data scientists and researchers to go beyond descriptive analysis and uncover deeper insights, allowing them to make data-driven decisions and formulate hypotheses about the broader context from which the data was sampled. 

3. Probability Distributions:

Probability distributions serve as foundational concepts in statistics and mathematics, providing a structured framework for characterizing the probabilities of various outcomes in random events. Poisson distributions offer structured representations for understanding how data is distributed across different values or occurrences.

Much like navigational charts guiding explorers through uncharted territory, probability distributions function as reliable guides through the landscape of uncertainty, enabling us to quantitatively assess the likelihood of specific events.

Read More —-> 7 types of statistical distributions with practical examples 

They constitute essential tools for statistical analysis, hypothesis testing, and predictive modeling, furnishing a systematic approach to evaluate, analyze, and make informed decisions in scenarios involving randomness and unpredictability. Comprehension of probability distributions is imperative for effectively modeling and interpreting real-world data and facilitating accurate predictions. 

4. Sampling Methods:

We now know inferential statistics help us make conclusions about the population from a sample of the population. How do we ensure that the sample is representative of the population? This is where sampling methods come to aid us.

Sampling methods are a set of methods that help us pick our sample set out of the population. Sampling methods are indispensable in surveys, experiments, and observational studies, ensuring that our conclusions are both efficient and statistically valid.

There are many types of sampling methods. Some of the most common ones are defined below. 

• Simple Random Sampling: A method where each member of the population has an equal chance of being selected for the sample, typically through random processes. 
• Stratified Sampling: The population is divided into subgroups (strata), and a random sample is taken from each stratum in proportion to its size. 
• Systematic Sampling: Selecting every “kth” element from a population list, using a systematic approach to create the sample. 
• Cluster Sampling: The population is divided into clusters, and a random sample of clusters is selected, with all members in selected clusters included. 

Understand Bootstrap Sampling

• Convenience Sampling: Selection of individuals/items based on convenience or availability, often leading to non-representative samples. 
• Purposive (Judgmental) Sampling: Researchers deliberately select specific individuals/items based on their expertise or judgment, potentially introducing bias. 
• Quota Sampling: The population is divided into subgroups, and individuals are purposively selected from each subgroup to meet predetermined quotas. 
• Snowball Sampling: Used in hard-to-reach populations, where participants refer researchers to others, leading to an expanding sample. 

5. Regression Analysis:

Regression analysis is a statistical method that helps us quantify the relationship between a dependent variable and one or more independent variables. It’s like drawing a line through data points to understand and predict how changes in one variable relate to changes in another.

Regression models, such as linear regression or logistic regression, are used to uncover patterns and causal relationships in diverse fields like economics, healthcare, and social sciences. This technique empowers researchers to make predictions, analyze cause-and-effect connections, and gain insights into complex phenomena. 

Unveil Rank-Based Encoding in Regression for Surefire Success

6. Hypothesis Testing:

Hypothesis testing is a key field of statistical concepts used to assess claims or hypotheses about a population using sample data. It’s like a process of weighing evidence to determine if there’s enough proof to support a hypothesis.

Researchers formulate a null hypothesis and an alternative hypothesis, then use statistical tests to evaluate whether the data supports rejecting the null hypothesis in favor of the alternative.

This method is crucial for making informed decisions, drawing meaningful conclusions, and assessing the significance of observed effects in various fields of research and decision-making. 

7. Data visualizations:

Data visualization is the art and science of representing complex data in a visual and comprehensible form. It’s like translating the language of numbers and statistics into a graphical story that anyone can understand at a glance.

Effective data visualization not only makes data more accessible but also allows us to spot trends, patterns, and outliers, making it an essential tool for data analysis and decision-making. Whether through charts, graphs, maps, or interactive dashboards, data visualization empowers us to convey insights, share information, and gain a deeper understanding of complex datasets. 

Check out some of the most important plots for Data Science here. 

8. ANOVA (Analysis of variance):

Analysis of Variance (ANOVA) is one of the statistical concepts used to compare the means of two or more groups to determine if there are significant differences among them. It’s like the referee in a sports tournament, checking if there’s enough evidence to conclude that the teams’ performances are different.

ANOVA calculates a test statistic and a p-value, which indicates whether the observed differences in means are statistically significant or likely occurred by chance.

This method is widely used in research and experimental studies, allowing researchers to assess the impact of different factors or treatments on a dependent variable and draw meaningful conclusions about group differences. ANOVA is a powerful tool for hypothesis testing and plays a vital role in various fields, from medicine and psychology to economics and engineering. 

9. Time Series analysis:

Time series analysis is a specialized field of statistical concepts and data science that focuses on studying data points collected, recorded, or measured over time. It’s like examining the historical trajectory of a variable to understand its patterns and trends.

Learn about time series in Python tutorials

Time series analysis involves techniques for data visualization, smoothing, forecasting, and modeling to uncover insights and make predictions about future values.

This discipline finds applications in various domains, from finance and economics to climate science and stock market predictions, helping analysts and researchers understand and harness the temporal patterns within their data. 

10. Bayesian Statistics:

Bayesian statistics is a branch of statistics that takes a unique approach to probability and inference. Unlike classical statistics, which use fixed parameters, Bayesian statistics treat probability as a measure of uncertainty, updating beliefs based on prior information and new evidence.

It’s like continually refining your knowledge as you gather more data. Bayesian methods are particularly useful when dealing with complex, uncertain, or small-sample data, and they have applications in fields like machine learning, Bayesian networks, and decision analysis.

Conclusion

Statistics is more than just numbers—it serves as the backbone of data science, enabling the extraction of insights, making predictions, and driving informed decisions. From descriptive measures to Bayesian analysis, each one of the statistical concepts plays a vital role in understanding and interpreting data effectively.

Mastering these principles equips data scientists with the tools to navigate uncertainty, validate hypotheses, and communicate findings clearly. As data continues to shape industries and innovations, a strong foundation in statistics remains essential for thriving in the data-driven world.

Probability distributions are fundamental to data science, influencing the ways in which we analyze and interpret data. They offer a systematic approach to modeling uncertainty, facilitating predictions, and extracting insights from real-world events.

Whether it’s understanding customer behavior or refining machine learning models, probability distributions are crucial in the decision-making process. In this blog, we will delve into nine fundamental probability distributions commonly used in data science

Whether you’re new to the field or a seasoned data scientist, gaining expertise in these distributions will significantly improve your data analysis skills. In the realm of data science, understanding probability distributions is crucial. They provide a mathematical framework for modeling and analyzing data.  

Understand the applications of probability in data science  

Explore Probability Distributions in Data Science with Practical Applications

Following are the nine important data science distributions and their practical applications:

1. Normal Distribution

The normal distribution, characterized by its bell-shaped curve, is prevalent in various natural phenomena. For instance, IQ scores in a population tend to follow a normal distribution. This allows psychologists and educators to understand the distribution of intelligence levels and make informed decisions regarding education programs and interventions.  

Learn how AI is empowering the education industry 

Heights of adult males in a given population often exhibit a normal distribution. In such a scenario, most men tend to cluster around the average height, with fewer individuals being exceptionally tall or short.

This means that the majority fall within one standard deviation of the mean, while a smaller percentage deviates further from the average. 

2. Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of Bernoulli trials. Imagine conducting 10 coin flips and counting the number of heads. This scenario follows a binomial distribution. In practice, this distribution is used in fields like manufacturing, where it helps in estimating the probability of defects in a batch of products. 

Imagine a basketball player with a 70% free throw success rate. If this player attempts 10 free throws, the number of successful shots follows a binomial distribution. This distribution allows us to calculate the probability of making a specific number of successful shots out of the total attempts. 

3. Bernoulli Distribution

The Bernoulli distribution models a random variable with two possible outcomes: success or failure. Consider a scenario where a coin is tossed. Here, the outcome can be either a head (success) or a tail (failure). This distribution finds application in various fields, including quality control, where it’s used to assess whether a product meets a specific quality standard. 

When flipping a fair coin, the outcome of each flip can be modeled using a Bernoulli distribution. This distribution is aptly suited as it accounts for only two possible results – heads or tails. The probability of success (getting a head) is 0.5, making it a fundamental model for simple binary events. 

4. Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time or space, assuming a constant rate. For example, in a call center, the number of calls received in an hour can often be modeled using a Poisson distribution. This information is crucial for optimizing staffing levels to meet customer demands efficiently. 

Learn about 5 tips to enhance customer service using data science

In the context of a call center, the number of incoming calls over a given period can often be modeled using a Poisson distribution. This distribution is applicable when events occur randomly and are relatively rare, like calls to a hotline or requests for customer service during specific hours. 

5. Exponential Distribution

The exponential distribution represents the time until a continuous, random event occurs. In the context of reliability engineering, this distribution is employed to model the lifespan of a device or system before it fails. This information aids in maintenance planning and ensuring uninterrupted operation. 

The time intervals between successive earthquakes in a certain region can be accurately modeled by an exponential distribution. This is especially true when these events occur randomly over time, but the probability of them happening in a particular time frame is constant. 

6. Gamma Distribution

The gamma distribution extends the concept of the exponential distribution to model the sum of k independent exponential random variables. This distribution is used in various domains, including queuing theory, where it helps in understanding waiting times in systems with multiple stages. 

Consider a scenario where customers arrive at a service point following a Poisson process, and the time it takes to serve them follows an exponential distribution. In this case, the total waiting time for a certain number of customers can be accurately described using a gamma distribution. This is particularly relevant for modeling queues and wait times in various service industries. 

7. Beta Distribution

The beta distribution is a continuous probability distribution bound between 0 and 1. It’s widely used in Bayesian statistics to model probabilities and proportions. In marketing, for instance, it can be applied to optimize conversion rates on a website, allowing businesses to make data-driven decisions to enhance user experience. 

In the realm of A/B testing, the conversion rate of users interacting with two different versions of a webpage or product is often modeled using a beta distribution. This distribution allows analysts to estimate the uncertainty associated with conversion rates and make informed decisions regarding which version to implement. 

8. Uniform Distribution

In a uniform distribution, all outcomes have an equal probability of occurring. A classic example is rolling a fair six-sided die. In simulations and games, the uniform distribution is used to model random events where each outcome is equally likely. 

When rolling a fair six-sided die, each outcome (1 through 6) has an equal probability of occurring. This characteristic makes it a prime example of a discrete uniform distribution, where each possible outcome has the same likelihood of happening. 

Get started with your data science learning journey with our instructor-led live bootcamp. Explore now.  

9. Log Normal Distribution

The log normal distribution describes a random variable whose logarithm is normally distributed. In finance, this distribution is applied to model the prices of financial assets, such as stocks. Understanding the log normal distribution is crucial for making informed investment decisions. 

The distribution of wealth among individuals in an economy often follows a log-normal distribution. This means that when the logarithm of wealth is considered, the resulting values tend to cluster around a central point, reflecting the skewed nature of wealth distribution in many societies.  

Learn Probability Distributions Today! 

Understanding these distributions and their applications empowers data scientists to make informed decisions and build accurate models. Remember, the choice of distribution greatly impacts the interpretation of results, so it’s a critical aspect of data analysis. 

Delve deeper into probability with this short tutorial 

Final Thoughts

Getting a solid grasp of probability distributions is key to making sense of data and creating reliable models in data science. Each type of distribution has its own unique role—whether it’s capturing natural patterns with the normal distribution or forecasting rare occurrences with the Poisson distribution.

By mastering these tools, data scientists can make smarter decisions, fine-tune algorithms, and improve real-world outcomes.

As you dive deeper into your data science journey, keep exploring these distributions and how they’re applied in practice. The more you understand them, the stronger and more impactful your data-driven insights will become!

4. Chaos Theory

Chaos theory is a branch of mathematics that studies the behavior of dynamic systems. It is used to model many different types of systems, such as the weather, the stock market, and the human heart.

One of the most important concepts in chaos theory is the butterfly effect. The butterfly effect states that small changes in the initial conditions of a system can lead to large changes in the long-term behavior of the system.

Example:

Suppose you have a butterfly flapping its wings in Brazil. This could cause a small change in the atmosphere, which could eventually lead to a hurricane in Florida.

Chaos theory is used in many different areas of physics, engineering, and economics. It is also used in everyday life, such as when predicting the weather and managing financial risks.

Learn about the Top 7 Statistical Techniques

Physics:

5. Law of Gravity

Formula: F = G * (m1 * m2) / r^2

The law of gravity is a physical law that describes the gravitational force between two objects. It states that the force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.

Example:

Suppose you have two objects, each with a mass of 1 kg. The gravitational force between the two objects would be 6.67 x 10^-11 N.

If you double the distance between the two objects, the gravitational force between them would be halved.

The law of gravity is used in many different areas of work, including astronomy, space exploration, and engineering. It is also used in everyday life, such as when calculating the weight of an object or the trajectory of a projectile.

Complex Numbers

6.The square root of minus one

Formula: i = sqrt(-1)

The square root of minus one is a complex number that is denoted by the letter i. It is defined as the number that, when multiplied by itself, equals -1.

Example:

i * i = -1

The square root of minus one is used in many different areas of mathematics, physics, and engineering. It is also used in everyday life, such as when calculating the voltage and current in an electrical circuit.

Read the Top 10 Statistics Books for Data Science

9. Information Theory

Formula: H(X) = -∑p(x) log2(p(x))

Information theory is a branch of mathematics that studies the transmission and processing of information. It was developed by Claude Shannon in the mid-20th century.

One of the most important concepts in information theory is entropy. Entropy is a measure of the uncertainty in a message. The higher the entropy of a message, the more uncertain it is.

Example:

Suppose you have a coin. The entropy of the coin is 1 bit, because there are two equally likely outcomes: heads or tails.

If you flip the coin and it lands on heads, the entropy of the coin is 0, because there is only one possible outcome: heads.

Information theory is used in many different areas of communication, computer science, and statistics. It is also used in everyday life, such as when designing data compression algorithms and communication protocols.

Another interesting article on probability distribution

Physics and Engineering:

Physics and Cosmology:

Finance and Economics:

17. Black-Scholes Equation

Formula: ∂C/∂t + ½σ^2S^2∂^2C/∂S^2 – rC = 0

The Black-Scholes equation is a differential equation that describes the price of a European option. A European option is a financial contract that gives the holder the right, but not the obligation, to buy or sell an asset at a certain price on a certain date.

The Black-Scholes equation is used to price options and to develop hedging strategies. It is one of the most important equations in finance.

Example:

Suppose you are buying a call option on a stock. The Black-Scholes equation can be used to calculate the price of the call option. This information can then be used to decide whether or not to buy the call option and to determine how much to pay for it.

The Black-Scholes equation is used by many different financial institutions, such as investment banks and hedge funds. It is also used by individual investors to make investment decisions.

Probability or statistical likelihood is a fundamental concept in data science. It provides a framework for understanding and analyzing uncertainty, which is an essential aspect of many real-world problems. In this blog, we will discuss the importance of probability in data science, its applications, and how it can be used to make data-driven decisions. 

What is Probability? 

It is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain. For example, the Statistical likelihood of rolling a six on a fair die is 1/6 or approximately 0.17. 

In data science, it is used to quantify the uncertainty associated with data. It helps data scientists to make informed decisions by providing a way to model and analyze the variability of data. It is also used to build models that can predict future events or outcomes based on past data. 

Applications of Probability in Data Science 

There are many applications of probability in data science, some of which are discussed below: 

1. Statistical Inference

Statistical inference allows data scientists to make conclusions about a population from a sample. Using probability, it helps quantify uncertainty in estimates and test hypotheses. For example, in market research, probability helps determine how accurately survey results reflect customer preferences, guiding decisions in marketing and policy.

2. Machine Learning

In machine learning, probability helps algorithms make predictions based on past data. Classification models, like Naive Bayes, use probability to predict the likelihood of a new observation belonging to a specific class. It is also used in regression and decision trees to handle uncertainty and identify patterns for predicting future outcomes.

3. Bayesian Analysis

Bayesian analysis uses statistical likelihood to update beliefs based on new data. Unlike traditional methods, it incorporates prior knowledge, refining predictions over time. In healthcare, for example, Bayesian models can adjust the probability of a patient having a condition as more test results come in, making real-time decision-making possible.

4. Risk Assessment

Probability is crucial in assessing risks in industries such as finance and healthcare. By estimating the likelihood of events like loan defaults or disease outbreaks, organizations can make informed decisions to mitigate potential losses. For example, insurance companies use probability to determine the risk of insuring a policyholder based on their health or lifestyle.

5. Quality Control

In manufacturing, it helps ensure product quality. Statistical process control (SPC) uses probability to monitor production and detect deviations from expected behavior. By modeling defect rates and variability, manufacturers can improve product consistency and reduce waste, ensuring high-quality output.

6. Anomaly Detection

Probability is key in identifying anomalies or outliers in data. By modeling normal behavior, data scientists can flag unusual patterns, such as fraudulent transactions or cybersecurity breaches. Anomaly detection also helps predict equipment failures, allowing organizations to address issues before they escalate.

Key Probability Distributions in Data Science

Distributions are foundational for modeling data and uncertainty in data science. Understanding different types of distributions is essential for selecting the right statistical tools and constructing effective models. Below are some of the most important types of distributions:

• Normal Distribution: Often referred to as the bell curve, this is one of the most commonly used models in statistical analysis. It is widely applied in hypothesis testing, machine learning, and various statistical procedures. The normal distribution is crucial for modeling continuous data that tends to cluster around a mean.

• Binomial Distribution: This distribution is valuable for scenarios involving binary outcomes, such as success or failure. For example, it can predict the likelihood of a customer purchasing a product, based on prior behavior or marketing interventions. It is particularly useful when there are a fixed number of trials and each trial has two possible outcomes.

• Poisson Distribution: Often used to model the frequency of rare events occurring over a specified time or space, the Poisson distribution is ideal for scenarios like predicting the number of customer complaints or system failures. It helps estimate how often events will happen based on their average rate of occurrence.

• Exponential Distribution: Typically applied in reliability analysis and queueing theory, this distribution models the time between events in a Poisson process. It is commonly used in fields like telecommunications to predict the time between customer arrivals in a service line or the expected lifespan of a machine.

Advanced Probability Concepts in Data Science

1. Conditional Probability

Conditional probability focuses on the likelihood of an event occurring given that another has already taken place. This is essential in data science for building more accurate models and making refined predictions.

For instance, in insurance, understanding the likelihood of an individual being able to pay premiums based on their financial obligations, such as a house loan, can help improve risk assessment models. By considering these interdependencies, data scientists can create more reliable predictions that adjust for existing conditions, resulting in better decision-making and more personalized services.

2. Rare Event Analysis

While rare events occur infrequently, they often hold great significance in large datasets. Identifying and understanding these low-probability occurrences is crucial in fields such as anomaly detection and fraud prevention.

For example, in financial transactions, detecting rare fraudulent activities, though infrequent, can have a major impact on a business. Techniques designed to analyze such events help highlight outliers and abnormal patterns that could indicate risks or opportunities that would otherwise go unnoticed. By focusing on these rare cases, organizations can proactively mitigate potential threats and optimize their processes.

How Probability Helps in Making Data-Driven Decisions 

It help data scientists to make data-driven decisions by providing a way to quantify the uncertainty associated with data. By using  to model and analyze data, data scientists can: 

• Estimate the likelihood of future events or outcomes based on past data. 
• Assess the risk associated with a particular decision or action. 
• Identify patterns and relationships in data. 
• Make predictions about future trends or behavior. 
• Evaluate the effectiveness of different strategies or interventions. 

Practical Applications of Probability

In data science and various industries, probability helps quantify uncertainty and guide decision-making. Below are two key applications of probability that play a significant role in predictive modeling and risk management.

1. Monte Carlo Simulations

Monte Carlo simulations use random sampling to estimate complex probabilities. By simulating many possible scenarios, these methods predict outcomes in uncertain environments.

They’re widely applied in finance to assess investment risks, in engineering for reliability studies, and in risk management for predicting adverse events. When exact solutions are difficult to calculate, Monte Carlo simulations provide valuable insights through simulation-based estimates.

2. Bayesian Networks

Bayesian networks are graphical models that show how variables are probabilistically related. They help decision-making by illustrating how uncertainty in one factor affects others.

As new data comes in, these networks update the probabilities to refine predictions. They’re commonly used in healthcare to predict disease likelihood or in machine learning to improve model accuracy, making them essential tools for adaptive, data-driven decision-making.

Bayes’ Theorem and its Relevance in Data Science 

Bayes’ theorem, also known as Bayes’ rule or Bayes’ law, is a fundamental concept in probability theory that has significant relevance in data science. It is named after Reverend Thomas Bayes, an 18th-century British statistician and theologian, who first formulated the theorem. 

At its core, Bayes’ theorem provides a way to calculate the probability of an event based on prior knowledge or information about related events. It is commonly used in statistical inference and decision-making, especially in cases where new data or evidence becomes available. 

The theorem is expressed mathematically as follows: 

P(A|B) = P(B|A) * P(A) / P(B) 

Where: 

• P(A|B) is the probability of event A occurring given that event B has occurred. 
• P(B|A) is the probability of event B occurring given that event A has occurred. 
• P(A) is the prior probability of event A occurring. 
• P(B) is the prior probability of event B occurring. 

In data science, Bayes’ theorem is used to update the probability of a hypothesis or belief in light of new evidence or data. This is done by multiplying the prior probability of the hypothesis by the likelihood of the new evidence given that hypothesis.

Master Naive Bayes for powerful data analysis. Read this blog to understand valuable insights from your data!

For example, let’s say we have a medical test that can detect a certain disease, and we know that the test has a 95% accuracy rate (i.e., it correctly identifies 95% of people with the disease and 5% of people without it). We also know that the prevalence of the disease in the population is 1%. If we administer the test to a person and they test positive, we can use Bayes’ theorem to calculate the probability that they actually have the disease. 

In conclusion, Bayes’ theorem is a powerful tool for probabilistic inference and decision-making in data science. Incorporating prior knowledge and updating it with new evidence, it enables more accurate and informed predictions and decisions. 

Common Mistakes to Avoid in Probability Analysis 

Probability analysis is an essential aspect of data science, providing a framework for making informed predictions and decisions based on uncertain events. However, even the most experienced data scientists can make mistakes when applying probability analysis to real-world problems. In this article, we’ll explore some common mistakes to avoid: 

1. Assuming Independence

One of the most common mistakes in probability analysis is assuming that events are independent when they are not. Independence means that the occurrence of one event does not affect the likelihood of the other event. However, in many real-world situations, this assumption is false.

For instance, in a medical study, one might assume that the likelihood of developing a certain condition is independent of factors like age, gender, or family history. In reality, these factors are often highly correlated and can significantly influence the probability of developing the condition.

Neglecting such dependencies can result in misleading conclusions, as the model will underestimate or overestimate the actual risk. To avoid this mistake, data scientists should carefully examine the relationships between variables and consider using methods that account for these dependencies, such as conditional probability or correlation analysis.

2. Misinterpreting Probability

A frequent mistake in probability analysis is the misinterpretation of what a specific probability value means. For example, a probability of 0.5 does not indicate certainty that an event will happen—it only signifies that there is an equal chance of the event happening or not happening. This is often misunderstood in the context of decision-making or risk assessment.

Misunderstanding this concept can lead to flawed decision-making, especially in high-stakes environments like finance or healthcare. Data scientists must ensure that they accurately communicate the true meaning of probability values to stakeholders and be aware of how probability values are used in real-world predictions.

3. Neglecting Sample Size

Sample size plays a crucial role in the accuracy and reliability of probability analysis. Small sample sizes can lead to statistical anomalies and skewed results, while excessively large sample sizes can be inefficient and unnecessarily resource-intensive.

For example, a small sample might not accurately represent the broader population, leading to overconfidence in predictions or misrepresentations of trends. Conversely, a very large sample size may not provide additional insights and can make analysis overly complex without yielding meaningful improvements in accuracy.

Choosing the right sample size is a delicate balance that depends on the context and the desired level of precision. Power analysis and consideration of effect size are tools that can help data scientists determine the optimal sample size for a given problem.

4. Confusing Correlation and Causation

A common pitfall in data analysis is confusing correlation with causation. While two events may appear to be correlated—meaning they occur together—it doesn’t necessarily mean that one causes the other.

For example, there might be a correlation between the number of ice cream sales and the number of drownings in summer, but this doesn’t mean that buying ice cream causes drowning incidents. The correlation is more likely due to a third factor, such as warmer weather.

This misunderstanding can lead to incorrect conclusions and flawed recommendations. Establishing causality requires more rigorous analysis, such as controlled experiments or the use of statistical techniques like regression analysis, instrumental variables, or randomized controlled trials. Causality is more difficult to prove and often requires a deep understanding of the underlying system or domain.

5. Ignoring Prior Knowledge

In Bayesian probability analysis, prior knowledge plays a vital role in shaping the probability of an event. The prior reflects what is already known or assumed about a system before observing the data. Failing to incorporate this prior information—or neglecting to update it with new evidence—can lead to inaccurate predictions and decisions.

For instance, when estimating the likelihood of a disease based on test results, prior knowledge about the prevalence of the disease in the population is essential. If prior knowledge is ignored, the posterior probability of the disease could be biased or misleading. Data scientists must recognize the importance of updating beliefs based on new data and use appropriate methods to integrate prior information effectively, such as Bayesian updating.

6. Overreliance on Models

While probabilistic models can be powerful tools for analysis and prediction, they are not foolproof. An overreliance on models can lead to false confidence in their predictions, especially if the assumptions behind the model are flawed or if the model is applied to an inappropriate context. Every model has limitations, and its accuracy depends on the quality of the data, the assumptions made, and the methods used to build it.

For example, a machine learning model trained on biased or incomplete data may produce biased predictions, or it may overfit to the training data, leading to poor generalization to new data. Data scientists should always be mindful of the assumptions underlying their models, validate their predictions, and consider using multiple models or techniques to cross-check results.

Additionally, it’s important to communicate the limitations of models to stakeholders and avoid presenting them as infallible tools.

Conclusion 

Statistical likelihood is a crucial tool for data scientists, providing a framework to quantify uncertainty and make informed decisions. By understanding and applying probabilistic methods, data scientists can build more accurate models, predict outcomes, and deliver reliable insights. As data-driven decision-making continues to grow across industries, the ability to effectively apply probabilities will be essential for success. Mastering this skill empowers data scientists to navigate uncertainty and drive impactful, data-backed results in a variety of fields.

The Poisson process is a popular method of counting random events that occur at a certain rate. It is commonly used in situations where the timing of events appears to be random, but the rate of occurrence is known. For example, the frequency of earthquakes in a specific region or the number of car accidents at a location can be modeled using the Poisson process. 

Understanding the Parameters of the Poisson Process

The Poisson process is defined by several key properties: 

• Events happen at a steady rate over time 

• The probability of an event happening in a short period of time is inversely proportional to the duration of the interval, and 
• Events take place independently of one another.  

Additionally, the Poisson distribution governs the number of events that take place during a specific period, and the rate parameter (which determines the mean and variance) is the only parameter that can be used to describe it. 

Mathematical Definition of the Poisson Process

To calculate the probability of a given number of events occurring in a Poisson process, the Poisson distribution formula is used: P(x) = (lambda^x * e^(-lambda)) / x! where lambda is the rate parameter and x! is the factorial of x. 

The Poisson process can be applied to a wide range of real-world situations, such as the arrival of customers at a store, the number of defects in a manufacturing process, the number of calls received by a call center, the number of accidents at a particular intersection, and the number of emails received by a person in a given time-period.  

It’s essential to keep in mind that the Poisson process is a stochastic process that counts the number of events that have occurred in a given interval of time, while the Poisson distribution is a discrete probability distribution that describes the likelihood of events with a Poisson process happening in a given time-period.

Real Scenarios To Use the Poisson Process

The Poisson process is a popular counting method used in situations where events occur at a certain rate but are actually random and without a certain structure. It is frequently used to model the occurrence of events over time, such as the number of faults in a manufacturing process or the arrival of customers at a store. Some examples of real-life situations where the Poisson process can be applied include: 

• The arrival of customers at a store or other business: The rate at which customers arrive at a store can be modeled using a Poisson process, with the rate parameter representing the average number of customers that arrive per unit of time. 
• The number of defects in a manufacturing process: The rate at which defects occur in a manufacturing process can be modeled using a Poisson process, with the rate parameter representing the average number of defects per unit of time. 
• The number of calls received by a call center: The rate at which calls are received by a call center can be modeled using a Poisson process, with the rate parameter representing the average number of calls per unit of time. 
• The number of accidents at a particular intersection: The rate at which accidents occur at a particular intersection can be modeled using a Poisson process, with the rate parameter representing the average number of accidents per unit of time. 
• The number of emails received by a person in a given time period: The rate at which emails are received by a person can be modeled using a Poisson process, with the rate parameter representing the average number of emails received per unit of time. 

It’s also used in other branches of probability and statistics, including the analysis of data from experiments involving a large number of trials and the study of queues. 

Poisson Process Variants: Non-Homogeneous Poisson Process (NHPP)

While the standard Poisson process assumes a constant rate (λ) over time, many real-world scenarios involve event rates that change dynamically. This leads to the Non-Homogeneous Poisson Process (NHPP), where the rate function λ(t) varies instead of staying fixed.

How NHPP Works

In an NHPP, the probability of an event occurring depends on time-varying intensity rather than a fixed average rate. Instead of using a constant λ, the process follows a function λ(t) that defines how the intensity changes over time.

The cumulative intensity function, called the mean value function Λ(t), is calculated as:

Λ(t)=∫0tλ(u) du\Lambda(t) = \int_0^t \lambda(u) \, duΛ(t)=∫0t​λ(u)du

This integral determines the expected number of events up to time t.

Applications of NHPP

1. Finance – NHPP models fluctuations in stock market transactions, where trading activity is higher during market openings and closings.
2. Healthcare – Used to model disease outbreaks where infection rates vary seasonally or due to interventions.
3. Weather Forecasting – Applied to rainfall modeling, where storm intensity fluctuates over time rather than occurring at a constant rate.
4. Network Traffic – Internet and telecommunications traffic often peaks during certain hours, making NHPP useful in modeling call volumes and data requests.

Simulating NHPP

NHPP can be simulated by dividing time into small intervals and adjusting the event probability based on λ(t). A common approach is thinning, where candidate event times from a standard Poisson process are filtered based on the varying rate function.

Why NHPP Matters

Unlike the traditional Poisson process, NHPP provides flexibility in modeling real-world situations where event occurrences are not uniform. It helps in better forecasting, risk analysis, and resource planning, making it a crucial tool in fields dealing with dynamic event rates.

Poisson Process in Machine Learning & AI

In machine learning and AI, many real-world problems involve predicting how often an event will occur within a fixed time or space. This is where the Poisson distribution becomes useful, as it models the probability of a given number of events happening in a fixed interval. From natural language processing (NLP) to recommendation systems, Poisson-based models help make sense of count-based data and improve predictive accuracy.

Poisson Distribution in NLP

One key application of the Poisson distribution is in text analysis, where it helps predict how often a word appears in a document. Rare words tend to follow a Poisson-like distribution, making this useful for:

• Topic Modeling – Identifying themes based on word frequency.
• Spam Detection – Recognizing unusual word patterns in spam emails.
• Speech Recognition – Estimating the occurrence of specific words in spoken language.

Predicting User Behavior in AI Systems

AI-powered systems often rely on Poisson processes to model event occurrences in various domains:

• Web Traffic Analysis – Predicting the number of visitors per hour on a website.
• Ad Click Predictions – Estimating how often users will click on an online ad.
• Customer Support Queues – Forecasting the number of service requests in a given period.

Poisson Regression for Predictive Modeling

Poisson regression is a valuable technique when dealing with count-based predictions, such as forecasting the number of hospital visits, sales transactions, or social media shares. Unlike linear regression, which assumes continuous outcomes, Poisson regression is tailored for non-negative integer outputs, making it ideal for applications in healthcare, risk assessment, and demand forecasting.

By incorporating the Poisson process into machine learning models, AI systems can better predict how often an event will happen, leading to more accurate and data-driven decision-making.

Limitations of the Poisson Process

The Poisson process is a useful tool for modeling random events, but it comes with several limitations that can affect its accuracy in real-world applications. Here are some key constraints to keep in mind:

• Independence of Events – Assumes events occur independently, but real-world events are often correlated (e.g., aftershocks following an earthquake).
• No Simultaneous Events – It does not allow multiple events to happen at the exact same time, which is unrealistic in many systems (e.g., network traffic congestion).
• Memoryless Property – Assumes the time until the next event is independent of past events, which is not true for systems where past occurrences influence future ones (e.g., machine wear and tear).
• Limited to Rare Events – Works best for rare events and may not be accurate for high-frequency occurrences (e.g., peak-hour internet traffic).
• Homogeneity in Space – Assumes events occur uniformly in time but does not account for spatial variability (e.g., earthquake epicenter distribution).
• Inability to Model Overdispersion – Assumes variance equals the mean, but many real-world datasets have greater variability (e.g., fluctuating insurance claims).
• Limited to Countable Events – Cannot model continuous data, making it unsuitable for non-discrete phenomena (e.g., rainfall intensity).
• Sensitivity to Time Intervals – Assumes a constant event rate, but real-world event rates often vary over time (e.g., hospital patient arrivals between shifts).
• No Accommodation for External Factors – Does not account for external influences affecting event rates (e.g., vaccination rates impacting disease spread).

Wrapping Up

The Poisson process is a powerful tool for modeling events that happen at a certain average rate but in a random manner. It is defined by a constant event rate, independence between events, and an inverse relationship between interval length and event probability.

The Poisson distribution helps calculate the probability of a given number of events occurring in a fixed time frame. This makes it useful for real-world applications like customer arrivals, manufacturing defects, call center activity, and traffic accidents.

In machine learning and AI, the Poisson process is key for count-based predictions. It helps model word frequencies in NLP, user interactions in recommendation systems, and web traffic trends. Poisson regression is widely used in healthcare, finance, and marketing to predict event occurrences.

By integrating the Poisson process into analytics and AI, we can better understand event patterns and make data-driven decisions. Whether in queueing systems, risk assessment, or demand forecasting, it remains a fundamental concept in probability and statistics.

Research requires various tools to collect, analyze and disseminate information effectively. Some essential research tools include search engines like Google Scholar, JSTOR, and PubMed, reference management software like Zotero, Mendeley, and EndNote, statistical analysis tools like SPSS, R, and Stata, writing tools like Microsoft Word and Grammarly, and data visualization tools like Tableau and Excel.  

1. Google Scholar – Google Scholar is a search engine for scholarly literature, including articles, theses, books, and conference papers.

2. JSTOR – JSTOR is a digital library of academic journals, books, and primary sources.

3.PubMed – PubMed is a free search engine accessing primarily the MEDLINE database of references and abstracts on life sciences and biomedical topics. 

4. Web of Science: Web of Science is a citation index that allows you to search for articles, conference proceedings, and books across various scientific disciplines. 

5. Scopus – Scopus citation database that covers scientific, technical, medical, and social sciences literature. 

6. Zotero: Zotero is a free, open-source citation management tool that helps you organize your research sources, create bibliographies, and collaborate with others.

7. Mendeley – Mendeley is a reference management software that allows you to organize and share your research papers and collaborate with others.

8. EndNote – EndNoted is a software tool for managing bibliographies, citations, and references on the Windows and macOS operating systems. 

9. RefWorks – RefWorks is a web-based reference management tool that allows you to create and organize a personal database of references and generate citations and bibliographies.

10. Evernote – Evernote is a digital notebook that allows you to capture and organize your research notes, web clippings, and documents.

11. SPSS – SPSS is a statistical software package used for data analysis, data mining, and forecasting.

12. R – R is a free, open-source software environment for statistical computing and graphics.

13. Stata – Stata is a statistical software package that provides a suite of applications for data management and statistical analysis.

Other helpful tools for collaboration and organization include NVivo, Slack, Zoom, and Microsoft Teams. With these tools, researchers can effectively find relevant literature, manage references, analyze data, write research papers, create visual representations of data, and collaborate with peers. 

14. Excel – Excel is spreadsheet software used for organizing, analyzing, and presenting data.

15. Tableau – Tableau is a data visualization software that allows you to create interactive visualizations and dashboards.

16. NVivo – Nviva is a software tool for qualitative research and data analysis.

17. Slack – Slack is a messaging platform for team communication and collaboration.

18. Zoom – Zoom is a video conferencing software that allows you to conduct virtual meetings and webinars.

19. Microsoft Teams – Microsoft Teams is a collaboration platform that allows you to chat, share files, and collaborate with your team.

20. Qualtrics – Qualtrics is an online survey platform that allows researchers to design and distribute surveys, collect and analyze data, and generate reports.

Maximize Accuracy and Efficiency With Research Tools

Research is a vital aspect of any academic discipline, and it is critical to have access to appropriate research tools to facilitate the research process. Researchers require access to various research tools and software to conduct research, analyze data, and report research findings. Some standard research tools researchers use include search engines, reference management software, statistical analysis tools, writing tools, and data visualization tools.

Specialized research tools are also available for researchers in specific fields, such as GIS software for geographers and geneticist gene sequence analysis tools. These tools help researchers organize data, collaborate with peers, and effectively present research findings.

It is crucial for researchers to choose the right tools for their research project, as these tools can significantly impact the accuracy and reliability of research findings.

Conclusion

Summing it up, researchers today have access to an array of essential research tools that can help simplify the research process. From data collection to analysis and presentation, these tools make research more accessible, efficient, and accurate. By leveraging these tools, researchers can improve their work and produce more high-quality research.

Written by Prasad D Wilagama

For example, when it comes to assessing the credibility of a witness, we need to know how reliable the person is and how likely it is that the person is lying. Similarly, when it comes to making predictions about the future, it is important to factor in not just the accuracy of the forecast but also whether it is credible. 

Counterfactual causal inference: 

Counterfactual causal inference is a statistical technique that is used to evaluate the causal significance of historical events. Exploring how historical events may have unfolded under small changes in circumstances allows us to assess the importance of factors that may have caused the event. This technique can be used in a wide range of fields such as economics, history, and social sciences. There are multiple ways of doing counterfactual inference, such as Bayesian Structural Modelling. 

Overparametrized models and regularization: 

Overparametrized models are models that have more parameters than the number of observations. These models are prone to overfitting and are not generalizable to new data. Regularization is a technique that is used to combat overfitting in overparametrized models. Regularization adds a penalty term to the loss function to discourage the model from fitting the noise in the data. Two common types of regularization are L1 and L2 regularization. 

Generic computation algorithms: 

Generic computation algorithms are a set of algorithms that can be applied to a wide range of problems. These algorithms are often used to solve optimization problems, such as gradient descent and conjugate gradient. They are also used in machine learning, such as support vector machines and k-means clustering. 

Robust inference: 

Robust inference is a technique that is used to make inferences that are not sensitive to outliers or extreme observations. This technique is often used in cases where the data is contaminated with errors or outliers. There are several robust statistical methods such as the median and the Huber M-estimator. 

Read about: Key statistical distributions with real life scenarios

Bootstrapping and simulation-based inference: 

Bootstrapping and simulation-based inference are techniques that are used to estimate the precision of sample statistics and to evaluate and compare models. Bootstrapping is a resampling technique that is used to estimate the sampling distribution of a statistic by resampling the data with replacement.

Simulation-based inference is a method that is used to estimate the sampling distribution of a statistic by generating many simulated samples from the model. 

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Multilevel models: 

Multilevel models are a class of models that are used to account for the hierarchical structure of data. These models are often used in fields such as education, sociology, and epidemiology. They are also known as hierarchical linear models, mixed-effects models, or random coefficient models. 

Adaptive decision analysis: 

Adaptive Decision Analysis is a statistical technique that is used to make decisions under uncertainty. It involves modeling the decision problem, simulating the outcomes of the decision and updating the decision based on the new information. This method is often used in fields such as finance, engineering, and healthcare. 

Which statistical techniques are most used by you? 

This article discusses most of the statistical methods that are used in quantitative fields. These are often used to infer causal relationships between variables. 

The primary goal of any statistical way is to infer causality from observational data. It is usually difficult to achieve this goal for two reasons. First, observational data may be noisy and contaminated by errors. Second, variables are often correlated. To correctly infer causality, it is necessary to model these correlations and to account for any biases and confounding factors. 

As statistical techniques are often implemented using specific software packages, the implementations of each method often differ. This article first briefly describes the papers and software packages that are used in the following sections. It then describes the most common statistical techniques and the best practices that are associated with each technique. 

In this blog, we are going to learn the differences and similarities between linear regression and logistic regression. 

Regression is a statistical technique used in the fields of finance, investing, and other disciplines that aim to establish the nature and strength of the relationship between a single dependent variable (often represented by Y) and a number of independent variables (known as independent variables). 

Forecasting and prediction both require regression analysis. There is a lot of overlap between this and machine learning. This statistical approach is employed in a variety of industries, including 

Financial: Understanding stock price trends, making price predictions, and assessing insurance risk.

Marketing: Analyze the success of marketing initiatives and project product pricing and sales. 

Manufacturing: Assess the relationships between the variables that define a better engine and its performance. 

Medicine: To produce generic medications for ailments, forecast the various medication combinations.

The most popular variation of this method is linear regression, which is also known as simple regression or ordinary least squares (OLS). Based on a line of best fit, linear regression determines the linear relationship between two variables.  

The slope of a straight line used to represent linear regression thus indicates how changing one variable effect changing another. In a linear regression connection, the value of one variable when the value of the other is zero is represented by the y-intercept. There are also non-linear regression models, although they are far more complicated.

Terminologies used in regression analysis

Outliers 

The term “outlier” refers to an observation in a dataset that has an extremely high or very low value in comparison to the other observations, i.e., it does not belong to the population.

Multicollinearity

The independent variables are said to be multicollinear when there is a strong correlation between them.

Heteroscedasticity

Heteroscedasticity refers to the non-constant fluctuation between the target variable and the independent variable.

Both under- and over-fit

Overfitting may result from the usage of extraneous explanatory variables. Overfitting occurs when our algorithm performs admirably on the training set but falls short on the test sets.

Linear regression

In simple terms, linear regression is used to find a relationship between two variables: a Dependent variable (y) and an independent variable (X) with the help of a straight line. It also makes predictions for continuous or numeric variables such as sales, salary, age, and product price and shows us how the value of the dependent variable changes with the change in the value of an independent variable.

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Let’s say we have a dataset available consisting of house areas in square meters and their respective prices.

As the change in area results in a change in price change of a house, we will put the area on the X-axis as an independent variable and the Price on the Y-axis as a dependent variable.

On the chart, these data points would appear as a scatter plot, a set of points that may or may not appear to be organized along any line.

Now using this data, we are required to predict the price of houses having the following areas: 500, 2000, and 3500.

After plotting these points, if a linear pattern is visible, sketch a straight line as the line of best fit.

The best-fit line we draw minimizes the distance between it and the observed data. Estimating this line is a key component of regression analysis that helps to infer the relationships between a dependent variable and an independent variable.

Measures for linear regression

To understand the amount of error that exists between different models in linear regression, we use metrics. Let’s discuss some of the evaluation measures for regression:

• Mean Absolute Error

Mean absolute error measures the absolute difference between the predicted and actual values of the model. This metric is the average prediction error. Lower MAE values indicate a better fit.

• Root Mean Squared Error

Root Mean Squared Error indicates how different the residuals are from zero. Residuals represent the difference between the observed and predicted value of the dependent variable.

• R-Squared Measure

R squared Measure is the standard deviation of the residuals. The plot of the residuals shows the distance of the data points from the regression line. The root mean squared error squares the residuals, averages the residuals, and takes the square root. RMSE measures the difference between the actual target from the predicted values.

Lower RMSE values indicate shorter distances from the actual data point to the line and therefore a better fit. RMSE uses the same units as the dependent value.

Logistic regression

Additionally, logistic models can modify raw data streams to produce characteristics for various AI and machine learning methods. In reality, one of the often employed machine learning techniques for binary classification issues, or problems with two class values, includes logistic regression. These problems include predictions like “this or that,” “yes or no,” and “A or B.”

Read about logistic regression in R in this blog

The probability of occurrences can also be estimated using logistic regression, which includes establishing a link between feature likelihood and outcome likelihood. In other words, it can be applied to categorization by building a model that links the number of hours of study to the likelihood that a student would pass or fail.

Comparison of linear regression and logistic regression

The primary distinction between logistic and linear regression is that the output of logistic regression is constant whereas the output of linear regression is continuousutilized.

The outcome, or dependent variable, in logistic regression, has just two possible values. However, the output of a linear regression is continuous, which means that there are an endless number of possible values for it.

When the response variable is categorical, such as yes/no, true/false, and pass/fail, logistic regression is utilized. When the response variable is continuous, like hours, height, or weight, linear regression is utilized.

Logistic regression and linear regression, for instance, can predict various outcomes depending on the information about the amount of time a student spends studying and the results of their exams.

Curve, a visual representation of linear and logistic regression

A straight line, often known as a regression line, is used to indicate linear regression. This line displays the expected score on “y” for each value of “x.” Additionally, the distance between the data points on the plot and the regression line reveals model flaws.

In contrast, an S-shaped curve is revealed using logistic regression. Here, the orientation and steepness of the curve are affected by changes in the regression coefficients. So, it follows that a positive slope yields an S-shaped curve, but a negative slope yields a Z-shaped curve.

Which one to use – Linear regression or logistic regression?

Regression analysis requires careful attention to the problem statement, which must be understood before proceeding. It seems sensible to apply linear regression if the problem description mentions forecasts. If binary classification is included in the issue statement, logistic regression should be used. Similarly, we must assess each of our regression models in light of the problem statement. 

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Statistical distributions help us understand a problem better by assigning a range of possible values to the variables, making them very useful in data science and machine learning. Here are 6 types of distributions with intuitive examples that often occur in real-life data. 

In statistics, a distribution is simply a way to understand how a set of data points are spread over some given range of values.  

For example, distribution takes place when the merchant and the producer agree to sell the product during a specific time frame. This form of distribution is exhibited by the agreement reached between Apple and AT&T to distribute their products in the United States. 

Types of statistical distributions 

There are several statistical distributions, each representing different types of data and serving different purposes. Here we will cover several commonly used distributions. 

1. Normal Distribution 
2. t-Distribution 
3. Binomial Distribution 
4. Poisson Distribution 
5. Uniform Distribution 

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1. Normal Distribution 

A normal distribution also known as “Gaussian Distribution” shows the probability density for a population of continuous data (for example height in cm for all NBA players). Also, it indicates the likelihood that any NBA player will have a particular height. Let’s say fewer players are much taller or shorter than usual; most are close to average height.  

The spread of the values in our population is measured using a metric called standard deviation. The Empirical Rule tells us that: 

• 68.3% of the values will fall between1 standard deviation above and below the mean 
• 95.5% of the values will fall between2 standard deviations above and below the mean 
• 99.7% of the values will fall between3 standard deviations above and below the mean 

Let’s assume that we know that the mean height of all players in the NBA is 200cm and the standard deviation is 7cm. If Le Bron James is 206 cm tall, what proportion of NBA players is he taller than? We can figure this out! LeBron is 6cm taller than the mean (206cm – 200cm). Since the standard deviation is 7cm, he is 0.86 standard deviations (6cm / 7cm) above the mean. 

Our value of 0.86 standard deviations is called the z-score. This shows that James is taller than 80.5% of players in the NBA!  

This can be converted to a percentile using the probability density function (or a look-up table) giving us our answer. A probability density function (PDF) defines the random variable’s probability of coming within a distinct range of values. 

2. t-distribution 

A t-distribution is symmetrical around the mean, like a normal distribution, and its breadth is determined by the variance of the data. A t-distribution is made for circumstances where the sample size is limited, but a normal distribution works with a population. With a smaller sample size, the t-distribution takes on a broader range to account for the increased level of uncertainty. 

The number of degrees of freedom, which is determined by dividing the sample size by one, determines the curve of a t-distribution. The t-distribution tends to resemble a normal distribution as sample size and degrees of freedom increase because a bigger sample size increases our confidence in estimating the underlying population statistics. 

For example, suppose we deal with the total number of apples sold by a shopkeeper in a month. In that case, we will use the normal distribution. Whereas, if we are dealing with the total amount of apples sold in a day, i.e., a smaller sample, we can use the t distribution. 

3. Binomial distribution 

A Binomial Distribution can look a lot like a normal distribution’s shape. The main difference is that instead of plotting continuous data, it plots a distribution of two possible discrete outcomes, for example, the results from flipping a coin. Imagine flipping a coin 10 times, and from those 10 flips, noting down how many were “Heads”. It could be any number between 1 and 10. Now imagine repeating that task 1,000 times. 

If the coin, we are using is indeed fair (not biased to heads or tails) then the distribution of outcomes should start to look at the plot above. In the vast majority of cases, we get 4, 5, or 6 “heads” from each set of 10 flips, and the likelihood of getting more extreme results is much rarer! 

4. Bernoulli distribution 

The Bernoulli Distribution is a special case of Binomial Distribution. It considers only two possible outcomes, success, and failure, true or false. It’s a really simple distribution, but worth knowing! In the example below we’re looking at the probability of rolling a 6 with a standard die.

If we roll a die many, many times, we should end up with a probability of rolling a 6, 1 out of every 6 times (or 16.7%) and thus a probability of not rolling a 6, in other words rolling a 1,2,3,4 or 5, 5 times out of 6 (or 83.3%) of the time! 

5. Discrete uniform distribution: All outcomes are equally likely 

Uniform distribution is represented by the function U(a, b), where a and b represent the starting and ending values, respectively. Like a discrete uniform distribution, there is a continuous uniform distribution for continuous variables.  

In statistics, uniform distribution refers to a statistical distribution in which all outcomes are equally likely. Consider rolling a six-sided die. You have an equal probability of obtaining all six numbers on your next roll, i.e., obtaining precisely one of 1, 2, 3, 4, 5, or 6, equaling a probability of 1/6, hence an example of a discrete uniform distribution. 

As a result, the uniform distribution graph contains bars of equal height representing each outcome. In our example, the height is a probability of 1/6 (0.166667). 

The drawbacks of this distribution are that it often provides us with no relevant information. Using our example of a rolling die, we get the expected value of 3.5, which gives us no accurate intuition since there is no such thing as half a number on a dice. Since all values are equally likely, it gives us no real predictive power. 

It is a distribution in which all events are equally likely to occur. Below, we’re looking at the results from rolling a die many, many times. We’re looking at which number we got on each roll and tallying these up. If we roll the die enough times (and the die is fair) we should end up with a completely uniform probability where the chance of getting any outcome is exactly the same 

6. Poisson distribution 

A Poisson Distribution is a discrete distribution similar to the Binomial Distribution (in that we’re plotting the probability of whole numbered outcomes) Unlike the other distributions we have seen however, this one is not symmetrical – it is instead bounded between 0 and infinity.  

For example, a cricket chirps two times in 7 seconds on average. We can use the Poisson distribution to determine the likelihood of it chirping five times in 15 seconds. A Poisson process is represented with the notation Po(λ), where λ represents the expected number of events that can take place in a period.

The expected value and variance of a Poisson process is λ. X represents the discrete random variable. A Poisson Distribution can be modeled using the following formula. 

The Poisson distribution describes the number of events or outcomes that occur during some fixed interval. Most commonly this is a time interval like in our example below where we are plotting the distribution of sales per hour in a shop. 

Conclusion: 

Data is an essential component of the data exploration and model development process. We can adjust our Machine Learning models to best match the problem if we can identify the pattern in the data distribution, which reduces the time to get to an accurate outcome.  

Indeed, specific Machine Learning models are built to perform best when certain distribution assumptions are met. Knowing which distributions, we’re dealing with may thus assist us in determining which models to apply. 

The Monte Carlo method is a technique for solving complex problems using probability and random numbers. Through repeated random sampling, Monte Carlo calculates the probabilities of multiple possible outcomes occurring in an uncertain process.  

Whenever you try to solve problems in the future, you make certain assumptions. For example, forecasting problems make certain assumptions like the cost of a particular item, the value of stocks, or electricity units used in the future. Since these problems try to predict an estimate of an unknown value based on historical data, there always exists inherent risk and uncertainty.  

The Monte Carlo simulation allows us to see all the possible outcomes of our decisions and assess risk, consequently allowing for better decision-making under uncertainty. 

This blog will walk through the famous Monty Hall problem, and how it can be solved using the Monte Carlo method using Python.  

Monty Hall problem 

In the Monty Hall problem, the TV show host Monty presents three doors to the participant. Behind one of the doors is a valuable prize like a car, while behind the others is a less valuable prize like a goat.  

Consider yourself to be one of the participants in the show. You choose one out of the three doors. Before opening your chosen door, Monty opens another door behind which would be one of the goats. Now you are left with two doors, behind one could be the car, and behind the other would be the other goat. 

Monty then gives you the option to either switch your answer to the other unopened door or stick to the original one.  

Is it in your favor to switch your answer to the other door? Well, probability says it is!  

Let’s see how: 

Initially, there are three unopen doors in front of you. The probability of the car being behind any of these doors is 1/3.  

Let’s say you decide to pick door #1 as the probability is the same (1/3) for each of these doors. In other words, the probability that the car is behind door #1 is 1/3, and the probability that it will be behind either door #2 or door #3 is 2/3. 

Monty is aware of the prize behind each door. He chooses to open door #3 and reveal a goat. He then asks you if you would like to either switch to door #2 or stick with door #1.  

To solve the problem, let’s switch to Python and apply the Monte Carlo simulation. 

Solving with Python 

Initialize the 3 prizes

Create python lists to store the probabilities after each game. We will play as many games as iterations input.  

Monte Carlo simulation 

Before starting the game, we randomize the prizes behind each door. One of the doors will have a car behind it, while the other two will have a goat each. When we play a large number of games, all possible permutations get covered of prize distributions, and door choices get covered.  

Below is the code that decides if your choice was correct or not, and if switching would’ve been the correct move.  

 After playing each game, the winning probabilities are updated and stored in the lists. When all games have been played, we return the final values of each of the lists, i.e., winning by switching your choice and winning by sticking to your choice.  

Get results

Enter your desired number of iterations (the higher the number, the more numbers of games will be played to approximate the probabilities). In the final step, plot your results.  

After running the simulation 1000 times, the probability that we win by always switching is 67.7%, and the probability that we win by always sticking to our choice is 32.3%. In other words, you will win approximately 2/3 times if you switch your door, and only 1/3 times if you stick to the original door. 

Therefore, according to the Monte Carlo simulation, we are confident that it works to our advantage to switch the door in this tricky game. 

Written by Aadam Nadeem

In this blog, we will introduce you to the highly rated data science statistics books on Amazon. As you read the blog, you will find 5 books for beginners and 5 books for advanced-level experts. We will discuss what’s covered in each book and how it helps you to scale up your data science career. 

Advanced statistics books for data science 

1. Naked Statistics: Stripping the Dread from the Data – By Charles Wheelan 

The book unfolds the underlying impact of statistics on our everyday life. It walks the readers through the power of data behind the news. 

Mr. Wheelan begins the book with the classic Monty Hall problem. It is a famous, seemingly paradoxical problem using Bayes’ theorem in conditional probability. Moving on, the book separates the important ideas from the arcane technical details that can get in the way. The second part of the book interprets the role of descriptive statistics in crafting a meaningful summary of the underlying phenomenon of data. 

Wheelan highlights the Gini Index to show how it represents the income distribution of the nation’s residents and is mostly used to measure inequality. The later part of the book clarifies key concepts such as correlation, inference, and regression analysis explaining how data is being manipulated in order to tackle thorny questions.

Wheelan’s concluding chapter is all about the amazing contribution that statistics will continue to make to solving the world’s most pressing problems, rather than a more reflective assessment of its strengths and weaknesses.  

2. Bayesian Methods For Hackers – Probabilistic Programming and Bayesian Inference, By Cameron Davidson-Pilon 

We mostly learn Bayesian inference through intensely complex mathematical analyses that are also supported by artificial examples. This book comprehends Bayesian inference through probabilistic programming with the powerful PyMC language and the closely related Python tools NumPy, SciPy, and Matplotlib. 

Davidson-Pilon focused on improving learners’ understanding of the motivations, applications, and challenges in Bayesian statistics and probabilistic programming. Moreover, this book brings a much-needed introduction to Bayesian methods targeted at practitioners.

Therefore, you can reap the most benefit from this book if you have a prior sound understanding of statistics. Knowing about prior and posterior probabilities will give an added advantage to the reader in building and training the first Bayesian model.    

Read this blog if you want to learn in detail about statistical distributions

The second part of the book introduces the probabilistic programming library for Python through a series of detailed examples and intuitive explanations, with recent core developments and the popularity of the scientific stack in Python, PyMC is likely to become a core component soon enough.

PyMC does have dependencies to run, namely NumPy and (optionally) SciPy. To not limit the user, the examples in this book will rely only on PyMC, NumPy, SciPy, and Matplotlib. This book is filled with examples, figures, and Python code that make it easy to get started solving actual problems.  

3. Practical Statistics for Data Scientists – By Peter Bruce and Andrew Bruce  

This book is most beneficial for readers that have some basic understanding of R programming language and statistics.  

The authors penned the important concepts to teach practical statistics in data science and covered data structures, datasets, random sampling, regression, descriptive statistics, probability, statistical experiments, and machine learning. The code is available in both Python and R. If an example code is offered with this book, you may use it in your programs and documentation.  

The book defines the first step in any data science project that is exploring the data or data exploration. Exploratory data analysis is a comparatively new area of statistics. Classical statistics focused almost exclusively on inference, a sometimes-complex set of procedures for drawing conclusions about large populations based on small samples.  

To apply the statistical concepts covered in this book, unstructured raw data must be processed and manipulated into a structured form—as it might emerge from a relational database—or be collected for a study.  

4. Advanced Engineering Mathematics by Erwin Kreyszig 

Advanced Engineering Mathematics is a textbook for advanced engineering and applied mathematics students. The book deals with the calculus of vectors, tensor and differential equations, partial differential equations, linear elasticity, nonlinear dynamics, chaos theory, and applications in engineering. 

Advanced Engineering Mathematics is a textbook that focuses on the practical aspects of mathematics. It is an excellent book for those who are interested in learning about engineering and its role in society.

The book is divided into five sections: Differential Equations, Integral Equations, Differential Mathematics, Calculus, and Probability Theory. It also provides a basic introduction to linear algebra and matrix theory. This book can be used by students who want to study at the graduate level or for those who want to become engineers or scientists. 

The text provides a self-contained introduction to advanced mathematical concepts and methods in applied mathematics. It covers topics such as integral calculus, partial differentiation, vector calculus and its applications to physics, Hamiltonian systems and their stability analysis, functional analysis, and classical mechanics and its applications to engineering problems. 

The book includes a large number of problems at the end of each chapter that help students develop their understanding of the material covered in the chapter. 

5. Computer Age Statistical Inference by Bradley Efron and Trevor Hastie 

Computer Age Statistical Inference is a book aimed at data scientists who are looking to learn about the theory behind machine learning and statistical inference. The authors have taken a unique approach in this book, as they have not only introduced many different topics but have also included a few examples of how these ideas can be applied in practice.

The book starts off with an introduction to statistical inference and then progresses through chapters on linear regression models, logistic regression models, statistical model selection, and variable selection. There are several appendices that provide additional information on topics such as confidence intervals and variable importance. This book is great for anyone looking for an introduction to machine learning or statistics. 

Computer Age Statistical Inference is a book that introduces students to the field of statistical inference in a modern computational setting. It covers topics such as Bayesian inference and nonparametric methods, which are essential for data science. In particular, this book focuses on Bayesian classification methods and their application to real-world problems.

It discusses how to develop models for continuous and discrete data, how to evaluate model performance, how to choose between parametric and nonparametric methods, how to incorporate prior distributions into your model, and much more. 

5 Beginner level statistics books for data science 

6. How to Lie with Statistics by Darrell Huff 

How to Lie with Statistics is one of the most influential books about statistical inference. It was first published in 1954 and has been translated into many languages. The book describes how to use statistics to make your most important decisions, like whether to buy a house, how much money to give to charity, and what kind of mortgage you should take out.

The book is intended for laymen, as it includes illustrations and some mathematical formulas. It’s full of interesting insights into how people can manipulate data to support their own agendas. 

The book is still relevant today because it describes how people use statistics in their daily lives. It gives an understanding of the types of questions that are asked and how they are answered by statistical methods. The book also explains why some results seem more reliable than others. 

The first half of the book discusses methods of making statistical claims (including how to make improper ones) and illustrates these using examples from real life. The second half provides a detailed explanation of the mathematics behind probability theory and statistics. 

A common criticism of the book is that it focuses too much on what statisticians do rather than why they do it. This is true — but that’s part of its appeal! 

 7. Head-first Statistics: A Brain-Friendly Guide Book by Dawn Griffiths  

If you are looking for a book that will help you understand the basics of statistics, then this is the perfect book for you. In this book, you will learn how to use data and make informed decisions based on your findings. You will also learn how to analyze data and draw conclusions from it. 

This book is ideal for those who have already completed a course in statistics or have studied it in college. Griffiths has given an overview of the different types of statistical tests used in everyday life and provides examples of how to use them effectively. 

The book starts off with an explanation of statistics, which includes topics such as sampling, probability, population and sample size, normal distribution and variation, confidence intervals, tests of hypotheses and correlation.  

After this section, the book goes into more advanced topics such as regression analysis, hypothesis testing etc. There are also some chapters on data mining techniques like clustering and classification etc. 

The author has explained each topic in detail for the readers who have little knowledge about statistics so they can follow along easily. The language used throughout this book is very clear and simple which makes it easy to understand even for beginners. 

8. Think Stats By Allen B. Downey 

Think Stats is a great book for students who want to learn more about statistics. The author, Allen Downey, uses simple examples and diagrams to explain the concepts behind each topic. This book is especially helpful for those who are new to mathematics or statistics because it is written in an easy-to-understand manner that even those with a high school degree can understand. 

The book begins with an introduction to basic counting, addition, subtraction, multiplication, and division. It then moves on to finding averages and making predictions about what will happen if one number changes. It also covers topics like randomness, sampling techniques, sampling distributions, and probability theory. 

The author uses real-world examples throughout the book so that readers can see how these concepts apply to their own lives. He also includes exercises at the end of each chapter so that readers can practice what they’ve learned before moving on to the next section of the book.

This makes Think Stats an excellent resource for anyone looking for tips on improving their math skills or just wanting to brush up on some statistical basics! 

9. An Introduction To Statistical Learning With Applications In R By Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani 

Statistical learning with applications in R is a guide to advanced statistical learning. It introduces modern machine learning techniques and their applications, including sequential decision-making, Gaussian mixture models, boosting, and genetic programming.

The book covers methods for supervised and unsupervised learning, as well as neural networks. The book also includes chapters on Bayesian statistics and deep learning. 

It begins with a discussion of correlation and regression analysis, followed by Bayesian inference using Markov chain Monte Carlo methods. The authors then discuss regularization techniques for regression models and introduce boosting algorithms.

This section concludes with an overview of neural networks and convolutional neural networks (CNNs). The remainder of the book deals with topics such as kernel methods, support vector machines (SVMs), regression trees (RTs), naive Bayes classifiers, Gaussian processes (GP), gradient ascent methods, and more. 

This statistics book is recommended to researchers willing to learn about statistical machine learning but do not have the necessary expertise in mathematics or programming languages 

10. Statistics in Plain English By Timothy C. Urdan 

Statistics in Plain English is a writing guide for students of statistics. Timothy in his book covered basic concepts with examples and guidance for using statistical techniques in the real world. The book includes a glossary of terms, exercises (with solutions), and web resources. 

The book begins by explaining the difference between descriptive statistics and inferential statistics, which are used to draw conclusions about data. It then covers basic vocabulary such as mean, median, mode, standard deviation, and range.  In Chapter 2, the author explains how to calculate sample sizes that are large enough to make accurate estimates.

In Chapters 3–5 he gives examples of how to use various kinds of data: census data on population density; survey data on attitudes toward various products; weather reports on temperature fluctuations; and sports scores from games played by teams over time periods ranging from minutes to seasons. He also shows how to use these data to estimate the parameters for models that explain behavior in these situations. 

The last 3 chapters define the use of frequency distributions to answer questions about probability distributions such as whether there’s a significant difference between two possible outcomes or whether there’s a trend in a set of numbers over time or space 

Which data science statistics books are you planning to get? 

Build upon your statistical concepts and successfully step into the world of data science. Analyze your knowledge and choose the most suitable book for your career to enhance your data science skills. If you have any more suggestions for statistics books for data science, please share them with us in the comments below.  

Logistic regression is one of the statistical techniques in machine learning used to form prediction models. It is one of the most popular classification algorithms mostly used for binary classification problems (problems with two class values, however, some variants may deal with multiple classes as well). It’s used for various research and industrial problems.

Therefore, it is essential to have a good grasp of logistic regression algorithms while learning data science. This tutorial is a sneak peek from many of Data Science Dojo’s hands-on exercises from their data science Bootcamp program, you will learn how logistic regression fits a dataset to make predictions, as well as when and why to use it.

In short, Logistic Regression is used when the dependent variable(target) is categorical. For example:

• To predict whether an email is spam (1) or not spam (0)
• Whether the tumor is malignant (1) or not (0)

Intro to Logistic Regression

It is named ‘Logistic Regression’ because its underlying technology is quite the same as Linear Regression. There are structural differences in how linear and logistic regression operate. Therefore, linear regression isn’t suitable to be used for classification problems. This link answers in detail why linear regression isn’t the right approach for classification.

Its name is derived from one of the core functions behind its implementation called the logistic function or the sigmoid function. It’s an S-shaped curve that can take any real-valued number and map it into a value between 0 and 1, but never exactly at those limits.

The hypothesis function of logistic regression can be seen below where the function g(z) is also shown.

The hypothesis for logistic regression now becomes:

Here θ (theta) is a vector of parameters that our model will calculate to fit our classifier.

After calculations from the above equations, the cost function is now as follows:

This tutorial will follow the format below to provide you with hands-on practice with Logistic Regression:

1. Importing Libraries
2. Importing Datasets
3. Exploratory Data Analysis
4. Feature Engineering
5. Pre-processing
6. Model Development
7. Prediction
8. Evaluation

The scenario

In this tutorial, we will be working with the Default of Credit Card Clients Data Set. This data set has 30000 rows and 24 columns. The data set could be used to estimate the probability of default payment by credit card clients using the data provided. These attributes are related to various details about a customer, his past payment information, and bill statements. It is hosted in Data Science Dojo’s repository.

Think of yourself as a lead data scientist employed at a large bank. You have been assigned to predict whether a particular customer will default on their payment next month or not. The result is an extremely valuable piece of information for the bank to make decisions regarding offering credit to its customers and could massively affect the bank’s revenue. Therefore, your task is very critical. You will learn to use logistic regression to solve this problem.

The dataset is a tricky one as it has a mix of categorical and continuous variables. Moreover, you will also get a chance to practice these concepts through short assignments given at the end of a few sub-modules. Feel free to change the parameters in the given methods once you have been through the entire notebook.

Download Exercise Files

1) Importing libraries

We’ll begin by importing the dependencies that we require. The following dependencies are popularly used for data-wrangling operations and visualizations. We would encourage you to have a look at their documentation.

library(knitr)
library(tidyverse)
library(ggplot2)
library(mice)
library(lattice)
library(reshape2)
#install.packages("DataExplorer") if the following package is not available
library(DataExplorer)

2) Importing Datasets

The dataset is available at Data Science Dojo’s repository in the following link. We’ll use the head method to view the first few rows.

## Need to fetch the excel file
path <- "https://code.datasciencedojo.com/datasciencedojo/datasets/raw/master/
Default%20of%20Credit%20Card%20Clients/default%20of%20credit%20card%20clients.csv"
data <- read.csv(file = path, header = TRUE)
head(data)

Since the header names are in the first row of the dataset, we’ll use the code below to first assign the headers to be the one from the first row and then delete the first row from the dataset. This way we will get our desired form.

colnames(data) <- as.character(unlist(data[1,]))
data = data[-1, ]
head(data)

To avoid any complications ahead, we’ll rename our target variable “default payment next month” to a name without spaces using the code below.

colnames(data)[colnames(data)=="default payment next month"] <- "default_payment"
head(data)

3) Exploratory data analysis

Data Exploration is one of the most significant portions of the machine-learning process. Clean data can ensure a notable increase in the accuracy of our model. No matter how powerful our model is, it cannot function well unless the data we provide has been thoroughly processed.

This step will briefly take you through this step and assist you in visualizing your data, finding the relation between variables, dealing with missing values and outliers, and assisting in getting some fundamental understanding of each variable we’ll use.

Moreover, this step will also enable us to figure out the most important attributes to feed our model and discard those that have no relevance.

We will start by using the dim function to print out the dimensionality of our data frame.

dim(data)

30000 25

The str method will allow us to know the data type of each variable. We’ll transform it to a numeric data type since it’ll be easier to use for our functions ahead.

str(data)

'data.frame':	30000 obs. of  25 variables:
 $ ID             : Factor w/ 30001 levels "1","10","100",..: 1 11112 22223 23335 24446 25557 26668 27779 28890 2 ...
 $ LIMIT_BAL      : Factor w/ 82 levels "10000","100000",..: 14 5 81 48 48 48 49 2 7 14 ...
 $ SEX            : Factor w/ 3 levels "1","2","SEX": 2 2 2 2 1 1 1 2 2 1 ...
 $ EDUCATION      : Factor w/ 8 levels "0","1","2","3",..: 3 3 3 3 3 2 2 3 4 4 ...
 $ MARRIAGE       : Factor w/ 5 levels "0","1","2","3",..: 2 3 3 2 2 3 3 3 2 3 ...
 $ AGE            : Factor w/ 57 levels "21","22","23",..: 4 6 14 17 37 17 9 3 8 15 ...
 $ PAY_0          : Factor w/ 12 levels "-1","-2","0",..: 5 1 3 3 1 3 3 3 3 2 ...
 $ PAY_2          : Factor w/ 12 levels "-1","-2","0",..: 5 5 3 3 3 3 3 1 3 2 ...
 $ PAY_3          : Factor w/ 12 levels "-1","-2","0",..: 1 3 3 3 1 3 3 1 5 2 ...
 $ PAY_4          : Factor w/ 12 levels "-1","-2","0",..: 1 3 3 3 3 3 3 3 3 2 ...
 $ PAY_5          : Factor w/ 11 levels "-1","-2","0",..: 2 3 3 3 3 3 3 3 3 1 ...
 $ PAY_6          : Factor w/ 11 levels "-1","-2","0",..: 2 4 3 3 3 3 3 1 3 1 ...
 $ BILL_AMT1      : Factor w/ 22724 levels "-1","-10","-100",..: 13345 10030 10924 15026 21268 18423 12835 1993 1518 307 ...
 $ BILL_AMT2      : Factor w/ 22347 levels "-1","-10","-100",..: 11404 5552 3482 15171 16961 17010 13627 12949 3530 348 ...
 $ BILL_AMT3      : Factor w/ 22027 levels "-1","-10","-100",..: 18440 9759 3105 15397 12421 16866 14184 17258 2072 365 ...
 $ BILL_AMT4      : Factor w/ 21549 levels "-1","-10","-100",..: 378 11833 3620 10318 7717 6809 16081 8147 2129 378 ...
 $ BILL_AMT5      : Factor w/ 21011 levels "-1","-10","-100",..: 385 11971 3950 10407 6477 6841 14580 76 1796 2638 ...
 $ BILL_AMT6      : Factor w/ 20605 levels "-1","-10","-100",..: 415 11339 4234 10458 6345 7002 14057 15748 12215 3230 ...
 $ PAY_AMT1       : Factor w/ 7944 levels "0","1","10","100",..: 1 1 1495 2416 2416 3160 5871 4578 4128 1 ...
 $ PAY_AMT2       : Factor w/ 7900 levels "0","1","10","100",..: 6671 5 1477 2536 4508 2142 4778 6189 1 1 ...
 $ PAY_AMT3       : Factor w/ 7519 levels "0","1","10","100",..: 1 5 5 646 6 6163 4292 1 4731 1 ...
 $ PAY_AMT4       : Factor w/ 6938 levels "0","1","10","100",..: 1 5 5 337 6620 5 2077 5286 5 813 ...
 $ PAY_AMT5       : Factor w/ 6898 levels "0","1","10","100",..: 1 1 5 263 5777 5 950 1502 5 408 ...
 $ PAY_AMT6       : Factor w/ 6940 levels "0","1","10","100",..: 1 2003 4751 5 5796 6293 963 1267 5 1 ...
 $ default_payment: Factor w/ 3 levels "0","1","default payment next month": 2 2 1 1 1 1 1 1 1 1 ...

data[, 1:25] <- sapply(data[, 1:25], as.character)

We have involved an intermediate step by converting our data to character first. We need to use as.character before as.numeric. This is because factors are stored internally as integers with a table to give the factor level labels. Just using as.numeric will only give the internal integer codes.

data[, 1:25] <- sapply(data[, 1:25], as.numeric)
str(data)

'data.frame':	30000 obs. of  25 variables:
 $ ID             : num  1 2 3 4 5 6 7 8 9 10 ...
 $ LIMIT_BAL      : num  20000 120000 90000 50000 50000 50000 500000 100000 140000 20000 ...
 $ SEX            : num  2 2 2 2 1 1 1 2 2 1 ...
 $ EDUCATION      : num  2 2 2 2 2 1 1 2 3 3 ...
 $ MARRIAGE       : num  1 2 2 1 1 2 2 2 1 2 ...
 $ AGE            : num  24 26 34 37 57 37 29 23 28 35 ...
 $ PAY_0          : num  2 -1 0 0 -1 0 0 0 0 -2 ...
 $ PAY_2          : num  2 2 0 0 0 0 0 -1 0 -2 ...
 $ PAY_3          : num  -1 0 0 0 -1 0 0 -1 2 -2 ...
 $ PAY_4          : num  -1 0 0 0 0 0 0 0 0 -2 ...
 $ PAY_5          : num  -2 0 0 0 0 0 0 0 0 -1 ...
 $ PAY_6          : num  -2 2 0 0 0 0 0 -1 0 -1 ...
 $ BILL_AMT1      : num  3913 2682 29239 46990 8617 ...
 $ BILL_AMT2      : num  3102 1725 14027 48233 5670 ...
 $ BILL_AMT3      : num  689 2682 13559 49291 35835 ...
 $ BILL_AMT4      : num  0 3272 14331 28314 20940 ...
 $ BILL_AMT5      : num  0 3455 14948 28959 19146 ..
 $ BILL_AMT6      : num  0 3261 15549 29547 19131 ...
 $ PAY_AMT1       : num  0 0 1518 2000 2000 ...
 $ PAY_AMT2       : num  689 1000 1500 2019 36681 ...
 $ PAY_AMT3       : num  0 1000 1000 1200 10000 657 38000 0 432 0 ...
 $ PAY_AMT4       : num  0 1000 1000 1100 9000 ...
 $ PAY_AMT5       : num  0 0 1000 1069 689 ...
 $ PAY_AMT6       : num  0 2000 5000 1000 679 ...
 $ default_payment: num  1 1 0 0 0 0 0 0 0 0 ...

When applied to a data frame, the summary() function is essentially applied to each column, and the results for all columns are shown together. For a continuous (numeric) variable like “age”, it returns the 5-number summary showing 5 descriptive statistics as these are numeric values.

summary(data)

       ID          LIMIT_BAL            SEX          EDUCATION    
 Min.   :    1   Min.   :  10000   Min.   :1.000   Min.   :0.000  
 1st Qu.: 7501   1st Qu.:  50000   1st Qu.:1.000   1st Qu.:1.000  
 Median :15000   Median : 140000   Median :2.000   Median :2.000  
 Mean   :15000   Mean   : 167484   Mean   :1.604   Mean   :1.853  
 3rd Qu.:22500   3rd Qu.: 240000   3rd Qu.:2.000   3rd Qu.:2.000  
 Max.   :30000   Max.   :1000000   Max.   :2.000   Max.   :6.000  
    MARRIAGE          AGE            PAY_0             PAY_2        
 Min.   :0.000   Min.   :21.00   Min.   :-2.0000   Min.   :-2.0000  
 1st Qu.:1.000   1st Qu.:28.00   1st Qu.:-1.0000   1st Qu.:-1.0000  
 Median :2.000   Median :34.00   Median : 0.0000   Median : 0.0000  
 Mean   :1.552   Mean   :35.49   Mean   :-0.0167   Mean   :-0.1338  
 3rd Qu.:2.000   3rd Qu.:41.00   3rd Qu.: 0.0000   3rd Qu.: 0.0000  
 Max.   :3.000   Max.   :79.00   Max.   : 8.0000   Max.   : 8.0000  
     PAY_3             PAY_4             PAY_5             PAY_6        
 Min.   :-2.0000   Min.   :-2.0000   Min.   :-2.0000   Min.   :-2.0000  
 1st Qu.:-1.0000   1st Qu.:-1.0000   1st Qu.:-1.0000   1st Qu.:-1.0000  
 Median : 0.0000   Median : 0.0000   Median : 0.0000   Median : 0.0000  
 Mean   :-0.1662   Mean   :-0.2207   Mean   :-0.2662   Mean   :-0.2911  
 3rd Qu.: 0.0000   3rd Qu.: 0.0000   3rd Qu.: 0.0000   3rd Qu.: 0.0000  
 Max.   : 8.0000   Max.   : 8.0000   Max.   : 8.0000   Max.   : 8.0000  
   BILL_AMT1         BILL_AMT2        BILL_AMT3         BILL_AMT4      
 Min.   :-165580   Min.   :-69777   Min.   :-157264   Min.   :-170000  
 1st Qu.:   3559   1st Qu.:  2985   1st Qu.:   2666   1st Qu.:   2327  
 Median :  22382   Median : 21200   Median :  20089   Median :  19052  
 Mean   :  51223   Mean   : 49179   Mean   :  47013   Mean   :  43263  
 3rd Qu.:  67091   3rd Qu.: 64006   3rd Qu.:  60165   3rd Qu.:  54506  
 Max.   : 964511   Max.   :983931   Max.   :1664089   Max.   : 891586  
   BILL_AMT5        BILL_AMT6          PAY_AMT1         PAY_AMT2      
 Min.   :-81334   Min.   :-339603   Min.   :     0   Min.   :      0  
 1st Qu.:  1763   1st Qu.:   1256   1st Qu.:  1000   1st Qu.:    833  
 Median : 18105   Median :  17071   Median :  2100   Median :   2009  
 Mean   : 40311   Mean   :  38872   Mean   :  5664   Mean   :   5921  
 3rd Qu.: 50191   3rd Qu.:  49198   3rd Qu.:  5006   3rd Qu.:   5000  
 Max.   :927171   Max.   : 961664   Max.   :873552   Max.   :1684259  
    PAY_AMT3         PAY_AMT4         PAY_AMT5           PAY_AMT6       
 Min.   :     0   Min.   :     0   Min.   :     0.0   Min.   :     0.0  
 1st Qu.:   390   1st Qu.:   296   1st Qu.:   252.5   1st Qu.:   117.8  
 Median :  1800   Median :  1500   Median :  1500.0   Median :  1500.0  
 Mean   :  5226   Mean   :  4826   Mean   :  4799.4   Mean   :  5215.5  
 3rd Qu.:  4505   3rd Qu.:  4013   3rd Qu.:  4031.5   3rd Qu.:  4000.0  
 Max.   :896040   Max.   :621000   Max.   :426529.0   Max.   :528666.0  
 default_payment 
 Min.   :0.0000  
 1st Qu.:0.0000  
 Median :0.0000  
 Mean   :0.2212  
 3rd Qu.:0.0000  
 Max.   :1.0000

Using the introduced method, we can get to know the basic information about the dataframe, including the number of missing values in each variable.

introduce(data)

As we can observe, there are no missing values in the dataframe.

The information in summary above gives a sense of the continuous and categorical features in our dataset. However, evaluating these details against the data description shows that categorical values such as EDUCATION and MARRIAGE have categories beyond those given in the data dictionary. We’ll find out these extra categories using the value_counts method.

count(data, vars = EDUCATION)

The data dictionary defines the following categories for EDUCATION: “Education (1 = graduate school; 2 = university; 3 = high school; 4 = others)”. However, we can also observe 0 along with numbers greater than 4, i.e. 5 and 6. Since we don’t have any further details about it, we can assume 0 to be someone with no educational experience and 0 along with 5 & 6 can be placed in others along with 4.

count(data, vars = MARRIAGE)

The data dictionary defines the following categories for MARRIAGE: “Marital status (1 = married; 2 = single; 3 = others)”. Since category 0 hasn’t been defined anywhere in the data dictionary, we can include it in the ‘others’ category marked as 3.

#replace 0's with NAN, replace others too
data$EDUCATION[data$EDUCATION == 0] <- 4
data$EDUCATION[data$EDUCATION == 5] <- 4
data$EDUCATION[data$EDUCATION == 6] <- 4
data$MARRIAGE[data$MARRIAGE == 0] <- 3

count(data, vars = MARRIAGE)
count(data, vars = EDUCATION)

We’ll now move on to a multi-variate analysis of our variables and draw a correlation heat map from the DataExplorer library. The heatmap will enable us to find out the correlation between each variable. We are more interested in finding out the correlation between our predictor attributes with the target attribute default payment next month. The color scheme depicts the strength of the correlation between the 2 variables.

This will be a simple way to quickly find out how much of an impact a variable has on our final outcome. There are other ways as well to figure this out.

plot_correlation(na.omit(data), maxcat = 5L)

We can observe the weak correlation of AGE, BILL_AMT1, BILL_AMT2, BILL_AMT3, BILL_AMT4, BILL_AMT5, and BILL_AMT6 with our target variable.

Now let’s have a univariate analysis of our variables. We’ll start with the categorical variables and have a quick check on the frequency of distribution of categories. The code below will allow us to observe the required graphs. We’ll first draw the distribution for all PAY variables.

plot_histogram(data)

We can make a few observations from the above histogram. The distribution above shows that nearly all PAY attributes are rightly skewed.

4) Feature engineering

This step can be more important than the actual model used because a machine learning algorithm only learns from the data we give it, and creating features that are relevant to a task is absolutely crucial.

Analyzing our data above, we’ve been able to note the extremely weak correlation of some variables with the final target variable. The following are the ones that have significantly low correlation values: AGE, BILL_AMT2, BILL_AMT3, BILL_AMT4, BILL_AMT5, BILL_AMT6.

#deleting columns

data_new <- select(data, -one_of('ID','AGE', 'BILL_AMT2',
       'BILL_AMT3','BILL_AMT4','BILL_AMT5','BILL_AMT6'))
head(data_new)

5) Pre-processing

Standardization is a transformation that centers the data by removing the mean value of each feature and then scaling it by dividing (non-constant) features by their standard deviation. After standardizing data the mean will be zero and the standard deviation one.

It is most suitable for techniques that assume a Gaussian distribution in the input variables and work better with rescaled data, such as linear regression, logistic regression, and linear discriminate analysis. If a feature has a variance that is orders of magnitude larger than others, it might dominate the objective function and make the estimator unable to learn from other features correctly as expected.

In the code below, we’ll use the scale method to transform our dataset using it.

data_new[, 1:17] <- scale(data_new[, 1:17])

head(data_new)

The next task we’ll do is to split the data for training and testing as we’ll use our test data to evaluate our model. We will now split our dataset into train and test. We’ll change it to 0.3. Therefore, 30% of the dataset is reserved for testing while the remaining is for training. By default, the dataset will also be shuffled before splitting.

#create a list of random number ranging from 1 to number of rows from actual data 
#and 70% of the data into training data  

data2 = sort(sample(nrow(data_new), nrow(data_new)*.7))

#creating training data set by selecting the output row values
train <- data_new[data2,]

#creating test data set by not selecting the output row values
test <- data_new[-data2,]

Let us print the dimensions of all these variables using the dim method. You can notice the 70-30% split.

dim(train)
dim(test)

21000 18

9000 18

6) Model development

We will now move on to the most important step of developing our logistic regression model. We have already fetched our machine learning model in the beginning. Now with a few lines of code, we’ll first create a logistic regression model which has been imported from sci-kit learn’s linear model package to our variable named model.

Following this, we’ll train our model using the fit method with X_train and y_train which contain 70% of our dataset. This will be a binary classification model.

## fit a logistic regression model with the training dataset
log.model <- glm(default_payment ~., data = train, family = binomial(link = "logit"))

summary(log.model)

Call:
glm(formula = default_payment ~ ., family = binomial(link = "logit"), 
    data = train)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.1171  -0.6998  -0.5473  -0.2946   3.4915  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -1.465097   0.019825 -73.900  < 2e-16 ***
LIMIT_BAL   -0.083475   0.023905  -3.492 0.000480 ***
SEX         -0.082986   0.017717  -4.684 2.81e-06 ***
EDUCATION   -0.059851   0.019178  -3.121 0.001803 ** 
MARRIAGE    -0.107322   0.018350  -5.849 4.95e-09 ***
PAY_0        0.661918   0.023605  28.041  < 2e-16 ***
PAY_2        0.069704   0.028842   2.417 0.015660 *  
PAY_3        0.090691   0.031982   2.836 0.004573 ** 
PAY_4        0.074336   0.034612   2.148 0.031738 *  
PAY_5        0.018469   0.036430   0.507 0.612178    
PAY_6        0.006314   0.030235   0.209 0.834584    
BILL_AMT1   -0.123582   0.023558  -5.246 1.56e-07 ***
PAY_AMT1    -0.136745   0.037549  -3.642 0.000271 ***
PAY_AMT2    -0.246634   0.056432  -4.370 1.24e-05 ***
PAY_AMT3    -0.014662   0.028012  -0.523 0.600677    
PAY_AMT4    -0.087782   0.031484  -2.788 0.005300 ** 
PAY_AMT5    -0.084533   0.030917  -2.734 0.006254 ** 
PAY_AMT6    -0.027355   0.025707  -1.064 0.287277    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 22176  on 20999  degrees of freedom
Residual deviance: 19535  on 20982  degrees of freedom
AIC: 19571

Number of Fisher Scoring iterations: 6

7) Prediction

Below we’ll use the prediction method to find out the predictions made by our Logistic Regression method. We will first store the predicted results in our y_pred variable and print the first 10 rows of our test data set. Following this we will print the predicted values of the corresponding rows and the original labels that were stored in y_test for comparison.

test[1:10,]

## to predict using logistic regression model, probablilities obtained
log.predictions <- predict(log.model, test, type="response")

## Look at probability output
head(log.predictions, 10)

2

0.539623162720197

7

0.232835137994762

10

0.25988780274953

11

0.0556716133560243

15

0.422481223473459

22

0.165384552048511

25

0.0494775267027534

26

0.238225423596718

31

0.248366972046479

37

0.111907725985513

Below we are going to assign our labels with the decision rule that if the prediction is greater than 0.5, assign it 1 else 0.

log.prediction.rd <- ifelse(log.predictions > 0.5, 1, 0)
head(log.prediction.rd, 10)

2

1

7

0

10

0

11

0

15

0

22

0

25

0

26

0

31

0

37

0

Evaluation

We’ll now discuss a few evaluation metrics to measure the performance of our machine-learning model here. This part has significant relevance since it will allow us to understand the most important characteristics that led to our model development.

We will output the confusion matrix. It is a handy presentation of the accuracy of a model with two or more classes.

The table presents predictions on the x-axis and accuracy outcomes on the y-axis. The cells of the table are the number of predictions made by a machine learning algorithm.

According to an article the entries in the confusion matrix have the following meaning in the context of our study:

[[a b][c d]]

• a is the number of correct predictions that an instance is negative,
• b is the number of incorrect predictions that an instance is positive,
• c is the number of incorrect predictions that an instance is negative, and
• d is the number of correct predictions that an instance is positive.

table(log.prediction.rd, test[,18])

                 
log.prediction.rd    0    1
                0 6832 1517
                1  170  481

We’ll write a simple function to print the accuracy below

accuracy <- table(log.prediction.rd, test[,18])
sum(diag(accuracy))/sum(accuracy)

0.812555555555556

Conclusion

This tutorial has given you a brief and concise overview of the Logistic Regression algorithm and all the steps involved in achieving better results from our model. This notebook has also highlighted a few methods related to Exploratory Data Analysis, Pre-processing, and Evaluation, however, there are several other methods that we would encourage you to explore on our blog or video tutorials.

If you want to take a deeper dive into several data science techniques. Join our 5-day hands-on Data Science Bootcamp preferred by working professionals, we cover the following topics:

• Fundamentals of Data Mining
• Machine Learning Fundamentals
• Introduction to R
• Introduction to Azure Machine Learning Studio
• Data Exploration, Visualization, and Feature Engineering
• Decision Tree Learning
• Ensemble Methods: Bagging, Boosting, and Random Forest
• Regression: Cost Functions, Gradient Descent, Regularization
• Unsupervised Learning
• Recommendation Systems
• Metrics and Methods for Evaluating Predictive Models
• Introduction to Online Experimentation and A/B Testing
• Fundamentals of Big Data Engineering
• Hadoop and Hive
• Message Queues and Real-time Analytics
• NoSQL Databases and HBase
• Hack Project: Creating a Real-time IoT Pipeline
• Naive Bayes
• Logistic Regression
• Times Series Forecasting

This post was originally sponsored on What’s The Big Data.

Every cook knows how to avoid Type I Error: just remove the batteries. Let’s also learn how to reduce the chances of Type II errors. 

Why type I and type II errors matter

A/B testing is an essential component of large-scale online services today. So essential, that every online business worth mentioning has been doing it for the last 10 years.

A/B testing is also used in email marketing by all major online retailers. The Obama for America data science team received a lot of press coverage for leveraging data science, especially A/B testing during the presidential campaign.

Here is an interesting article on this topic along with a data science bootcamp that teaches a/b testing and statistical analysis.

If you have been involved in anything related to A/B testing (online experimentation) on UI, relevance or email marketing, chances are that you have heard of Type i and Type ii error. The usage of these terms is common but a good understanding of them is not.

I have seen illustrations as simple as this.

Examples of type I and type II errors

I intend to share two great examples I recently read that will help you remember this especially important concept in hypothesis testing.

Type I error: An alarm without a fire.

Type II error: A fire without an alarm.

Every cook knows how to avoid Type I Error – just remove the batteries. Unfortunately, this increases the incidences of Type II error.

Reducing the chances of Type II error would mean making the alarm hypersensitive, which in turn would increase the chances of Type I error.

Another way to remember this is by recalling the story of the Boy Who Cried Wolf.

Null hypothesis testing: There is no wolf.

Alternative hypothesis testing: There is a wolf.

Villagers believing the boy when there was no wolf (Reject the null hypothesis incorrectly): Type 1 Error. Villagers not believing the boy when there was a wolf (Rejecting alternative hypothesis incorrectly): Type 2 Error

Tailpiece

The purpose of the post is not to explain type 1 and type 2 error. If this is the first time you are hearing about these terms, here is the Wikipedia entry: Type I and Type II Error.

During political campaigns, political candidates not only need votes but also need financial contributions. Here’s what the campaign finance data set from 2012 looks like.

Understanding individual political contribution by the occupation of top 1% vs bottom 99% in political campaigns

A political candidate not only needs votes, but they also need money.  In today’s multi-media world millions of dollars are necessary to effectively run campaign finances.   To win the election battle citizens will be bombarded with ads that cost millions.  Other mounting expenses include wages for staff, consultants, surveyors, grassroots activists, media experts, wonks, and policy analysts.  The figures are staggering with the next presidential election year campaigns likely to cost more than ten billion dollars.

Opensecrets.org has summarized the money spent by presidential candidates, Senate and House candidates, political parties, and independent interest groups that played an influential role in the federal elections by cycle.  There’s no sign of less spending in future elections.

The 2016 presidential election cycle is already underway, the fund-raising war has already begun.  Koch brothers’ political organization released an $889 million budget in January 2015 supporting conservative campaigns in the 2016 presidential contest.  As for primary presidential candidates, the Hillary Clinton Campaign aims to raise at least $100 million for the primary election.  On the other side of the political aisle, analysts speculated primary candidate Jeb Bush will raise over $100 million when he discloses his financial position in July.

In my mind I imagine that money coming from millionaires and billionaires or mega-corporations intent on promoting candidates that favor their cause.  But who are these people?  And how about middle-class citizens like me?  Does my paltry $200 amount to anything?  Does the spending power of the 99% have any impact on the outcome of an election?   Even as a novice I knew I would never understand American politics by listening to TV talking heads or the candidates and their say-nothing ads but by following the money.

By investigating real data about where the stream of money dominating our elections comes from and the role it plays in the success of an election, I hope to find some insight into all the political noise.   Thanks to the Federal Election Campaign Act, which requires candidate committees, party committees, and political action committees (PACs) to disclose reports on the money they raise and spend and identify individuals who give more than $200 in an election cycle, a wealth of public data exists to explore. I choose to focus on individual contributions to federal committees greater than $200 for the election cycle 2011-2012.

The data is publicly available at http://www.fec.gov/finance/disclosure/ftpdet.shtml.

Creating the groups

In the 2012 election cycle, which includes congressional and primary elections, the total amount of individual donations collected was USD 784 million.  USD 220 million came from the top 1% of donors, which made up 28% of the total contribution.  These elites’ wealthy donors were 7119 individuals, each having donated at least USD 10,000 to federal committees.  So, who are the top 1%?  What do they do for a living that gave them such financial power to support political committees?

The unique occupation titles from the dataset are simply overwhelming and difficult to construct appropriate analysis. Thus, these occupations were classified into twenty-two occupation groups according to the employment definition from the Bureau of Statistics.  Additional categories were created due to a lack of definition to classify them into appropriate groups.  Among them are “Retired,” “Unemployed,” “Homemaker,” and “Politicians.”

Immediately from Figure 1, we observe the “Management” occupation group contributed the highest total amount in the 2012 cycle for Democrats, Republicans, and Other parties, respectively.  Other top donors by occupation groups are “Business and Financial Operations,” “Retired,” “Homemaker,” “Politicians,” and “Legal.”  Overall, the Republicans Parties received a more individual contribution from most of the occupation groups, with noticeable exception from “Legal” and “Arts, Design, Entertainment, Sports and Media.”  The total contribution given to “Other” non-Democratic/Republican was abysmal in comparison.

Figure 1: Total contribution of top 1% by occupation group

One might conclude that the reason for the “Management” group being the top donor is obvious given these people are CEOs, CFOs, Presidents, Directors, Managers, and many other management titles in a company.  According to the Bureau of Statistics, the “Management” group earned the highest median wages among all other occupation groups.  They simply had more to give.  The same argument could be applied to the “Business and Financial Operations” group, which is comprised of people who held jobs as investors, business owners, real estate developers, bankers, etc.

Perhaps we could look at the individual contribution by occupation group from another angle.  When analyzing the average contribution by occupation group, the “Politicians” group became top of the chart. Individuals belonging to this category are either currently holding public office or they had declared candidacy for office with no other occupation reported.  Since there is no limit on how much candidates may contribute to their committee, this group represents rich individuals funding their campaigns.

Figure 2: Average contribution of Top 1% by occupation groups

Suspiciously, the average amount per politician given to Republican committees is dramatically higher than other parties.  Further analysis indicated that the outlier is candidate John Huntsman, who donated about USD 5 million to his committee, “Jon Huntsman for President Inc.” This has inflated the average contribution dramatically.  The same phenomenon was also observed among the “Management” group, where the average contribution to the “Other” party was significantly higher compared to traditional parties.

Out of the five donors who contributed to an independent party from the “Management” group, William Bloomfield alone donated USD 1.3 million (out of the USD 1.45 million total amount collected) to his “Bloomfield for Congress” committee. According to the data, he was the Chairman of Baron Real Estate.  This is an example of a wealthy elite spending a hefty sum of money to buy his way into the election race.

Donald Trump, a billionaire business mogul made headlines recently by declaring his intention to run for presidency 2016 election.  He certainly has no trouble paying for his campaign. After excluding the occupation groups “Politicians” and “Management,” with the intention to visualize the comparison among groups more clearly, the contrast became less dramatic.  No doubt, the average contribution to Republicans Committees is consistently higher than other parties in most of the occupation groups.

Figure 3: Average contribution of Top 1% by occupation group excluding politicians and management group

Could a similar story of the top 1% be told for the bottom 99%?  Overall, the top 5 contributors by occupation group are quite similar between the top 1% and bottom 99%.  Once again “Management” group collectively raised the most amount of donations to the Democrats and Republicans Parties.  The biggest difference here is that “Politicians” is no longer the top contributor in the bottom 99% demographic.

Figure 4: Total contribution of bottom 99% by Occupation Group

Homemakers consistently rank high in both total contributions as well as average contribution, in both the top 1% and bottom 99%.  On average, homemakers from the bottom 99% donated about $1500 meanwhile homemakers from the top 1% donated about $30,000 to their chosen political committees.  Clearly across all levels of socioeconomic status spouses and stay-at-home parents play a key role in the fundraising war.  Since the term “Homemaker” is not well-defined, I can only assume their source of money comes from a spouse, inherited wealth, or personal savings.

Figure 5: Average contribution of bottom 99% by occupation group

Another observation we could draw from the average contribution from the 99% plot is that “Other” non-Democrats/Republicans Parties depend heavily on the 99% as a source of funding for their political campaigns. Third-party candidates appear to be drawing most of their support from the little guy.

Figure 6: Median wages and median contribution by occupation group

Median wages and median contribution

Another interesting question warranting further investigation is how the amount individuals contributed to political committees is proportionately consistent across occupation groups.  When we plotted median wages per occupation group side by side with median political contribution, the median of donation per group is rather constant while the median income varies significantly across groups.  This implies that despite contributing the most overall, as a percentage of their income the wealthiest donors contributed the least.

Campaign finance data: The takeaway

The take-home message from this analysis is that the top 1% wealthy elite seems to be driving the momentum of fundraising for the election campaign.  I suspect most of them have full intention to support candidates who would look out for their interest if indeed they got elected.  We middle-class citizens may not have the ability to compete financially with these millionaires and billionaires, but our single vote is as powerful as their vote.  The best thing we could do as citizens is to educate ourselves on issues that matter to the future of our country.

Learn more about data science here

Written by Jasmine Wilkerson